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Mathematics LibreTexts

10.E: Foundations of Trigonometry (Exercises)

10.1: Angles and their Measure

\subsection{Exercises}

In Exercises \ref{dmsfirst} - \ref{dmslast}, convert the angles into the DMS system. Round each of your answers to the nearest second.

\begin{multicols}{4}

\begin{enumerate}

\item $63.75^{\circ}$ \label{dmsfirst}

\item $200.325^{\circ}$

\item $-317.06^{\circ}$

\item $179.999^{\circ}$ \label{dmslast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

In Exercises \ref{decimaldegfirst} - \ref{decimaldeglast}, convert the angles into decimal degrees. Round each of your answers to three decimal places.

\begin{multicols}{4}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $125^{\circ} 50'$ \label{decimaldegfirst}

\item $-32^{\circ} 10' 12''$

\item $502^{\circ} 35'$

\item $237^{\circ} 58' 43''$ \label{decimaldeglast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

In Exercises \ref{orientedanglefirst} - \ref{orientedanglelast}, graph the oriented angle in standard position. Classify each angle according to where its terminal side lies and then give two coterminal angles, one of which is positive and the other negative.

\begin{multicols}{4}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $330^{\circ}$ \label{orientedanglefirst}

\item $-135^{\circ}$

\item $120^{\circ}$

\item $405^{\circ}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{4}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $-270^{\circ}$ \vphantom{$\dfrac{11\pi}{6}$}

\item $\dfrac{5\pi}{6}$

\item $-\dfrac{11\pi}{3}$

\item $\dfrac{5\pi}{4}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{4}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\dfrac{3\pi}{4}$

\item $-\dfrac{\pi}{3}$ \vphantom{$\dfrac{11\pi}{6}$}

\item $\dfrac{7\pi}{2}$

\item $\dfrac{\pi}{4}$ \vphantom{$\dfrac{11\pi}{6}$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{4}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $-\dfrac{\pi}{2}$ \vphantom{$\dfrac{11\pi}{6}$}

\item $\dfrac{7\pi}{6}$

\item $-\dfrac{5\pi}{3}$

\item $3\pi$ \vphantom{$\dfrac{11\pi}{6}$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{4}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $-2\pi$ \vphantom{$\dfrac{11\pi}{6}$}

\item $-\dfrac{\pi}{4}$ \vphantom{$\dfrac{11\pi}{6}$}

\item $\dfrac{15\pi}{4}$

\item $-\dfrac{13\pi}{6}$ \label{orientedanglelast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

In Exercises \ref{degreetoradianfirst} - \ref{degreetoradianlast}, convert the angle from degree measure into radian measure, giving the exact value in terms of $\pi$.

\begin{multicols}{4}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $0^{\circ}$ \label{degreetoradianfirst}

\item $240^{\circ}$

\item $135^{\circ}$

\item $-270^{\circ}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{4}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $-315^{\circ}$

\item $150^{\circ}$

\item $45^{\circ}$

\item $-225^{\circ}$ \label{degreetoradianlast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

In Exercises \ref{radiantodegreefirst} - \ref{radiantodegreelast}, convert the angle from radian measure into degree measure.

\begin{multicols}{4}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\pi$ \vphantom{$\dfrac{11\pi}{6}$} \label{radiantodegreefirst}

\item $-\dfrac{2\pi}{3}$

\item $\dfrac{7\pi}{6}$

\item $\dfrac{11\pi}{6}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{4}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\dfrac{\pi}{3}$ \vphantom{$\dfrac{11\pi}{6}$}

\item $\dfrac{5\pi}{3}$

\item $-\dfrac{\pi}{6}$ \vphantom{$\dfrac{11\pi}{6}$}

\item $\dfrac{\pi}{2}$ \vphantom{$\dfrac{11\pi}{6}$} \label{radiantodegreelast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\pagebreak

In Exercises \ref{orientedarcfirst} - \ref{orientedarclast}, sketch the oriented arc on the Unit Circle which corresponds to the given real number.

\begin{multicols}{5}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $t=\frac{5 \pi}{6}$ \label{orientedarcfirst}

\item $t=-\pi$

\item $t = 6$

\item $t = -2$

\item $t = 12$ \label{orientedarclast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \label{spinningyoyo} A yo-yo which is 2.25 inches in diameter spins at a rate of 4500 revolutions per minute. How fast is the edge of the yo-yo spinning in miles per hour? Round your answer to two decimal places.

\item How many revolutions per minute would the yo-yo in exercise \ref{spinningyoyo} have to complete if the edge of the yo-yo is to be spinning at a rate of 42 miles per hour? Round your answer to two decimal places.

\item \label{yoyotrick} In the yo-yo trick `Around the World,' the performer throws the yo-yo so it sweeps out a vertical circle whose radius is the yo-yo string. If the yo-yo string is 28 inches long and the yo-yo takes 3 seconds to complete one revolution of the circle, compute the speed of the yo-yo in miles per hour. Round your answer to two decimal places.

\item A computer hard drive contains a circular disk with diameter 2.5 inches and spins at a rate of 7200 RPM (revolutions per minute). Find the linear speed of a point on the edge of the disk in miles per hour. \label{harddrive}

\item A rock got stuck in the tread of my tire and when I was driving 70 miles per hour, the rock came loose and hit the inside of the wheel well of the car. How fast, in miles per hour, was the rock traveling when it came out of the tread? (The tire has a diameter of 23 inches.)

\item The Giant Wheel at Cedar Point is a circle with diameter 128 feet which sits on an 8 foot tall platform making its overall height is 136 feet. (Remember this from Exercise \ref{giantwheelcircle} in Section \ref{Circles}?) It completes two revolutions in 2 minutes and 7 seconds.\footnote{Source: \href{http://www.cedarpoint.com/public/par...nderline{Cedar Point's webpage}}.} Assuming the riders are at the edge of the circle, how fast are they traveling in miles per hour?

\label{giantwheelmotion}

\item Consider the circle of radius $r$ pictured below with central angle $\theta$, measured in radians, and subtended arc of length $s$. Prove that the area of the shaded sector is $A = \frac{1}{2} r^{2} \theta$.

(Hint: Use the proportion $\frac{A}{\text{area of the circle}} = \frac{s}{\text{circumference of the circle}}$.)

\label{circularsectorarea}

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\end{center}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

In Exercises \ref{sectorfirst} - \ref{sectorlast}, use the result of Exercise \ref{circularsectorarea} to compute the areas of the circular sectors with the given central angles and radii.

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\theta = \dfrac{\pi}{6}, \; r = 12$ \vphantom{$\dfrac{5\pi}{4}$} \label{sectorfirst}

\item $\theta = \dfrac{5\pi}{4}, \; r = 100$

\item $\theta = 330^{\circ}, \; r = 9.3$ \vphantom{$\dfrac{5\pi}{4}$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\theta =\pi, \; r = 1$

\item $\theta = 240^{\circ}, \; r = 5$

\item $\theta = 1^{\circ}, \; r = 117$ \label{sectorlast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item Imagine a rope tied around the Earth at the equator. Show that you need to add only $2\pi$ feet of length to the rope in order to lift it one foot above the ground around the entire equator. (You do NOT need to know the radius of the Earth to show this.)

\item With the help of your classmates, look for a proof that $\pi$ is indeed a constant.

\end{enumerate}

\newpage

\subsection{Answers}

\begin{multicols}{4}

\begin{enumerate}

\item $63^{\circ} 45'$

\item $200^{\circ} 19' 30''$

\item $-317^{\circ} 3' 36''$

\item $179^{\circ} 59' 56''$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{4}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $125.833^{\circ}$

\item $-32.17^{\circ}$

\item $502.583^{\circ}$

\item $237.979^{\circ}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2} \raggedcolumns

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $330^{\circ}$ is a Quadrant IV angle\\

coterminal with $690^{\circ}$ and $-30^{\circ}$

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\item $-135^{\circ}$ is a Quadrant III angle\\

coterminal with $225^{\circ}$ and $-495^{\circ}$

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\begin{multicols}{2} \raggedcolumns

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $120^{\circ}$ is a Quadrant II angle\\

coterminal with $480^{\circ}$ and $-240^{\circ}$

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\item $405^{\circ}$ is a Quadrant I angle\\

coterminal with $45^{\circ}$ and $-315^{\circ}$

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%\pagebreak

\begin{multicols}{2} \raggedcolumns

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $-270^{\circ}$ \vphantom{$\dfrac{5\pi}{6}$} lies on the positive $y$-axis\\

coterminal with $90^{\circ}$ and $-630^{\circ}$ \vphantom{$\dfrac{17\pi}{6}$}

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\item $\dfrac{5\pi}{6}$ is a Quadrant II angle\\

coterminal with $\dfrac{17\pi}{6}$ and $-\dfrac{7\pi}{6}$

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\begin{enumerate}

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\item $-\dfrac{11\pi}{3}$ \vphantom{$\dfrac{17\pi}{6}$} is a Quadrant I angle\\

coterminal with $\dfrac{\pi}{3}$ and $-\dfrac{5\pi}{3}$ \vphantom{$\dfrac{17\pi}{4}$}

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\item $\dfrac{5\pi}{4}$ is a Quadrant III angle\\

coterminal with $\dfrac{13\pi}{4}$ and $-\dfrac{3\pi}{4}$

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\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\dfrac{3\pi}{4}$ is a Quadrant II angle\\

coterminal with $\dfrac{11\pi}{4}$ and $-\dfrac{5\pi}{4}$

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\item $-\dfrac{\pi}{3}$ \vphantom{$\dfrac{17\pi}{4}$} is a Quadrant IV angle\\

coterminal with $\dfrac{5\pi}{3}$ and $-\dfrac{7\pi}{3}$ \vphantom{$\dfrac{17\pi}{4}$}

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\item $\dfrac{7\pi}{2}$ lies on the negative $y$-axis \vphantom{$\dfrac{17\pi}{4}$}\\

coterminal with $\dfrac{3\pi}{2}$ and $-\dfrac{\pi}{2}$ \vphantom{$\dfrac{17\pi}{4}$}

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\arrow \reverse \polyline{(0, -5), (0,0), (5,0)}

\point[3pt]{(0,0)}

\arrow \parafcn{0,625,5}{(t+100)*dir(t)/400}

\tlpointsep{5pt}

\scriptsize

\axislabels {x}{{$-4 \hspace{7pt}$} -4, {$-3 \hspace{7pt}$} -3, {$-2 \hspace{7pt}$} -2, {$-1 \hspace{7pt}$} -1, {$1$} 1, {$2$} 2, {$3$} 3, {$4$} 4}

\axislabels {y}{{$-1$} -1, {$-2$} -2, {$-3$} -3, {$-4$} -4, {$1$} 1, {$2$} 2, {$3$} 3, {$4$} 4}

\normalsize

\end{mfpic}

\item $\dfrac{\pi}{4}$ is a Quadrant I angle \vphantom{$\dfrac{17\pi}{2}$}\\

coterminal with $\dfrac{9 \pi}{4}$ and $-\dfrac{7\pi}{4}$

\begin{mfpic}[12]{-5}{5}{-5}{5}

\drawcolor[gray]{0.7}

\axes

\xmarks{-4,-3,-2,-1,1,2,3,4}

\ymarks{-4,-3,-2,-1,1,2,3,4}

\tlabel(5,-0.5){\scriptsize $x$}

\tlabel(0.25,4.75){\scriptsize $y$}

\drawcolor[rgb]{0.33,0.33,0.33}

\arrow \reverse \polyline{(3.5355, 3.5355), (0,0), (5,0)}

\point[3pt]{(0,0)}

\arrow \arc[c]{(0,0), (2.5, 0.1), 40}

\tlpointsep{5pt}

\scriptsize

\axislabels {x}{{$-4 \hspace{7pt}$} -4, {$-3 \hspace{7pt}$} -3, {$-2 \hspace{7pt}$} -2, {$-1 \hspace{7pt}$} -1, {$1$} 1, {$2$} 2, {$3$} 3, {$4$} 4}

\axislabels {y}{{$-1$} -1, {$-2$} -2, {$-3$} -3, {$-4$} -4, {$1$} 1, {$2$} 2, {$3$} 3, {$4$} 4}

\normalsize

\end{mfpic}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2} \raggedcolumns

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $-\dfrac{\pi}{2}$ lies on the negative $y$-axis \vphantom{$\dfrac{17\pi}{4}$}\\

coterminal with $\dfrac{3\pi}{2}$ and $-\dfrac{5\pi}{2}$

\begin{mfpic}[12]{-5}{5}{-5}{5}

\drawcolor[gray]{0.7}

\axes

\xmarks{-4,-3,-2,-1,1,2,3,4}

\ymarks{-4,-3,-2,-1,1,2,3,4}

\tlabel(5,-0.5){\scriptsize $x$}

\tlabel(0.25,4.75){\scriptsize $y$}

\drawcolor[rgb]{0.33,0.33,0.33}

\arrow \reverse \polyline{(0, -5), (0,0), (5,0)}

\point[3pt]{(0,0)}

\arrow \arc[c]{(0,0), (2.5, -0.1), -85}

\tlpointsep{5pt}

\scriptsize

\axislabels {x}{{$-4 \hspace{7pt}$} -4, {$-3 \hspace{7pt}$} -3, {$-2 \hspace{7pt}$} -2, {$-1 \hspace{7pt}$} -1, {$1$} 1, {$2$} 2, {$3$} 3, {$4$} 4}

\axislabels {y}{{$-1$} -1, {$-2$} -2, {$-3$} -3, {$-4$} -4, {$1$} 1, {$2$} 2, {$3$} 3, {$4$} 4}

\normalsize

\end{mfpic}

\item $\dfrac{7\pi}{6}$ is a Quadrant III angle\\

coterminal with $\dfrac{19 \pi}{6}$ and $-\dfrac{5\pi}{6}$

\begin{mfpic}[12]{-5}{5}{-5}{5}

\drawcolor[gray]{0.7}

\axes

\xmarks{-4,-3,-2,-1,1,2,3,4}

\ymarks{-4,-3,-2,-1,1,2,3,4}

\tlabel(5,-0.5){\scriptsize $x$}

\tlabel(0.25,4.75){\scriptsize $y$}

\drawcolor[rgb]{0.33,0.33,0.33}

\arrow \reverse \polyline{(-4.3301,-2.5), (0,0), (5,0)}

\point[3pt]{(0,0)}

\arrow \arc[c]{(0,0), (2.5, 0.1), 205}

\tlpointsep{5pt}

\scriptsize

\axislabels {x}{{$-4 \hspace{7pt}$} -4, {$-3 \hspace{7pt}$} -3, {$-2 \hspace{7pt}$} -2, {$-1 \hspace{7pt}$} -1, {$1$} 1, {$2$} 2, {$3$} 3, {$4$} 4}

\axislabels {y}{{$-1$} -1, {$-2$} -2, {$-3$} -3, {$-4$} -4, {$1$} 1, {$2$} 2, {$3$} 3, {$4$} 4}

\normalsize

\end{mfpic}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2} \raggedcolumns

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $-\dfrac{5\pi}{3}$ is a Quadrant I angle \vphantom{$\dfrac{17\pi}{4}$} \\

coterminal with $\dfrac{\pi}{3}$ and $-\dfrac{11\pi}{3}$ \vphantom{$\dfrac{17\pi}{4}$}

\begin{mfpic}[12]{-5}{5}{-5}{5}

\drawcolor[gray]{0.7}

\axes

\xmarks{-4,-3,-2,-1,1,2,3,4}

\ymarks{-4,-3,-2,-1,1,2,3,4}

\tlabel(5,-0.5){\scriptsize $x$}

\tlabel(0.25,4.75){\scriptsize $y$}

\drawcolor[rgb]{0.33,0.33,0.33}

\arrow \reverse \polyline{(2.5, 4.3301), (0,0), (5,0)}

\point[3pt]{(0,0)}

\arrow \arc[c]{(0,0), (2.5, -0.1), -295}

\tlpointsep{5pt}

\scriptsize

\axislabels {x}{{$-4 \hspace{7pt}$} -4, {$-3 \hspace{7pt}$} -3, {$-2 \hspace{7pt}$} -2, {$-1 \hspace{7pt}$} -1, {$1$} 1, {$2$} 2, {$3$} 3, {$4$} 4}

\axislabels {y}{{$-1$} -1, {$-2$} -2, {$-3$} -3, {$-4$} -4, {$1$} 1, {$2$} 2, {$3$} 3, {$4$} 4}

\normalsize

\end{mfpic}

\item $3\pi$ lies on the negative $x$-axis \vphantom{$\dfrac{17\pi}{4}$} \\

coterminal with $\pi$ and $-\pi$ \vphantom{$\dfrac{17\pi}{4}$}

\begin{mfpic}[12]{-5}{5}{-5}{5}

\drawcolor[gray]{0.7}

\axes

\xmarks{-4,-3,-2,-1,1,2,3,4}

\ymarks{-4,-3,-2,-1,1,2,3,4}

\tlabel(5,-0.5){\scriptsize $x$}

\tlabel(0.25,4.75){\scriptsize $y$}

\drawcolor[rgb]{0.33,0.33,0.33}

\arrow \reverse \polyline{(-5,0), (0,0), (5,0)}

\point[3pt]{(0,0)}

\arrow \parafcn{0,535,5}{(t+100)*dir(t)/400}

\tlpointsep{5pt}

\scriptsize

\axislabels {x}{{$-4 \hspace{7pt}$} -4, {$-3 \hspace{7pt}$} -3, {$-2 \hspace{7pt}$} -2, {$-1 \hspace{7pt}$} -1, {$1$} 1, {$2$} 2, {$3$} 3, {$4$} 4}

\axislabels {y}{{$-1$} -1, {$-2$} -2, {$-3$} -3, {$-4$} -4, {$1$} 1, {$2$} 2, {$3$} 3, {$4$} 4}

\normalsize

\end{mfpic}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2} \raggedcolumns

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $-2\pi$ lies on the positive $x$-axis \vphantom{$\dfrac{17\pi}{4}$} \\

coterminal with $2\pi$ and $-4\pi$ \vphantom{$\dfrac{17\pi}{4}$}

\begin{mfpic}[12]{-5}{5}{-5}{5}

\drawcolor[gray]{0.7}

\axes

\xmarks{-4,-3,-2,-1,1,2,3,4}

\ymarks{-4,-3,-2,-1,1,2,3,4}

\tlabel(5,-0.5){\scriptsize $x$}

\tlabel(0.25,4.75){\scriptsize $y$}

\drawcolor[rgb]{0.33,0.33,0.33}

\point[3pt]{(0,0)}

\polyline{(0,0), (5,0)}

\arrow \parafcn{0,355,5}{(t+100)*dir(0-t)/300}

\tlpointsep{5pt}

\scriptsize

\axislabels {x}{{$-4 \hspace{7pt}$} -4, {$-3 \hspace{7pt}$} -3, {$-2 \hspace{7pt}$} -2, {$-1 \hspace{7pt}$} -1, {$1$} 1, {$2$} 2, {$3$} 3, {$4$} 4}

\axislabels {y}{{$-1$} -1, {$-2$} -2, {$-3$} -3, {$-4$} -4, {$1$} 1, {$2$} 2, {$3$} 3, {$4$} 4}

\normalsize

\end{mfpic}

\item $-\dfrac{\pi}{4}$ is a Quadrant IV angle \vphantom{$\dfrac{17\pi}{4}$} \\

coterminal with $\dfrac{7 \pi}{4}$ and $-\dfrac{9\pi}{4}$

\begin{mfpic}[12]{-5}{5}{-5}{5}

\drawcolor[gray]{0.7}

\axes

\xmarks{-4,-3,-2,-1,1,2,3,4}

\ymarks{-4,-3,-2,-1,1,2,3,4}

\tlabel(5,-0.5){\scriptsize $x$}

\tlabel(0.25,4.75){\scriptsize $y$}

\drawcolor[rgb]{0.33,0.33,0.33}

\arrow \reverse \polyline{(3.5355,-3.5355), (0,0), (5,0)}

\point[3pt]{(0,0)}

\arrow \arc[c]{(0,0), (2.5, -0.1), -40}

\tlpointsep{5pt}

\scriptsize

\axislabels {x}{{$-4 \hspace{7pt}$} -4, {$-3 \hspace{7pt}$} -3, {$-2 \hspace{7pt}$} -2, {$-1 \hspace{7pt}$} -1, {$1$} 1, {$2$} 2, {$3$} 3, {$4$} 4}

\axislabels {y}{{$-1$} -1, {$-2$} -2, {$-3$} -3, {$-4$} -4, {$1$} 1, {$2$} 2, {$3$} 3, {$4$} 4}

\normalsize

\end{mfpic}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2} \raggedcolumns

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\dfrac{15\pi}{4}$ is a Quadrant IV angle\\

coterminal with $\dfrac{7\pi}{4}$ and $-\dfrac{\pi}{4}$

\begin{mfpic}[12]{-5}{5}{-5}{5}

\drawcolor[gray]{0.7}

\axes

\xmarks{-4,-3,-2,-1,1,2,3,4}

\ymarks{-4,-3,-2,-1,1,2,3,4}

\tlabel(5,-0.5){\scriptsize $x$}

\tlabel(0.25,4.75){\scriptsize $y$}

\drawcolor[rgb]{0.33,0.33,0.33}

\arrow \reverse \polyline{(3.5355,-3.5355), (0,0), (5,0)}

\point[3pt]{(0,0)}

\arrow \parafcn{0,670,5}{(t+100)*dir(t)/400}

\tlpointsep{5pt}

\scriptsize

\axislabels {x}{{$-4 \hspace{7pt}$} -4, {$-3 \hspace{7pt}$} -3, {$-2 \hspace{7pt}$} -2, {$-1 \hspace{7pt}$} -1, {$1$} 1, {$2$} 2, {$3$} 3, {$4$} 4}

\axislabels {y}{{$-1$} -1, {$-2$} -2, {$-3$} -3, {$-4$} -4, {$1$} 1, {$2$} 2, {$3$} 3, {$4$} 4}

\normalsize

\end{mfpic}

\item $-\dfrac{13\pi}{6}$ is a Quadrant IV angle\\

coterminal with $\dfrac{11\pi}{6}$ and $-\dfrac{\pi}{6}$

\begin{mfpic}[12]{-5}{5}{-5}{5}

\drawcolor[gray]{0.7}

\axes

\xmarks{-4,-3,-2,-1,1,2,3,4}

\ymarks{-4,-3,-2,-1,1,2,3,4}

\tlabel(5,-0.5){\scriptsize $x$}

\tlabel(0.25,4.75){\scriptsize $y$}

\drawcolor[rgb]{0.33,0.33,0.33}

\arrow \reverse \polyline{(4.3301,-2.5), (0,0), (5,0)}

\point[3pt]{(0,0)}

\arrow \parafcn{0,385,5}{(t+100)*dir(0-t)/300}

\tlpointsep{5pt}

\scriptsize

\axislabels {x}{{$-4 \hspace{7pt}$} -4, {$-3 \hspace{7pt}$} -3, {$-2 \hspace{7pt}$} -2, {$-1 \hspace{7pt}$} -1, {$1$} 1, {$2$} 2, {$3$} 3, {$4$} 4}

\axislabels {y}{{$-1$} -1, {$-2$} -2, {$-3$} -3, {$-4$} -4, {$1$} 1, {$2$} 2, {$3$} 3, {$4$} 4}

\normalsize

\end{mfpic}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{4}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $0$ \vphantom{$\dfrac{17\pi}{4}$}

\item $\dfrac{4\pi}{3}$

\item $\dfrac{3\pi}{4}$

\item $-\dfrac{3\pi}{2}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{4}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $-\dfrac{7\pi}{4}$

\item $\dfrac{5\pi}{6}$

\item $\dfrac{\pi}{4}$ \vphantom{$\dfrac{17\pi}{4}$}

\item $-\dfrac{5\pi}{4}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{4}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $180^{\circ}$

\item $-120^{\circ}$

\item $210^{\circ}$

\item $330^{\circ}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{4}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $60^{\circ}$

\item $300^{\circ}$

\item $-30^{\circ}$

\item $90^{\circ}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2} \raggedcolumns

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $t = \dfrac{5\pi}{6}$

\begin{mfpic}[10]{-5}{5}{-5}{5}

\axes

\tlabel(5,-0.5){\scriptsize $x$}

\tlabel(0.5,5){\scriptsize $y$}

\tlabel(3.1,-0.75){\scriptsize $1$}

\tlabel(0.25,3.1){\scriptsize $1$}

\xmarks{-3 step 3 until 3}

\ymarks{-3 step 3 until 3}

\dotted \polyline{(0,0), (-4.3301,2.5)}

\drawcolor[gray]{0.7}

\circle{(0,0),3}

\drawcolor[rgb]{0.33,0.33,0.33}

\penwd{1.5pt}

\arrow \parafcn{0, 150, 5}{3*dir(t)}

\end{mfpic}

\item $t = -\pi$ \vphantom{$\dfrac{5\pi}{6}$}

\begin{mfpic}[10]{-5}{5}{-5}{5}

\axes

\tlabel(5,-0.5){\scriptsize $x$}

\tlabel(0.5,5){\scriptsize $y$}

\tlabel(3.1,-0.75){\scriptsize $1$}

\tlabel(0.25,3.1){\scriptsize $1$}

\xmarks{-3 step 3 until 3}

\ymarks{-3 step 3 until 3}

\drawcolor[gray]{0.7}

\circle{(0,0),3}

\drawcolor[rgb]{0.33,0.33,0.33}

\penwd{1.5pt}

\arrow \parafcn{0, 180, 5}{3*dir(-t)}

\end{mfpic}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2} \raggedcolumns

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $t = 6$

\begin{mfpic}[10]{-5}{5}{-5}{5}

\axes

\tlabel(5,-0.5){\scriptsize $x$}

\tlabel(0.5,5){\scriptsize $y$}

\tlabel(3.1,-0.75){\scriptsize $1$}

\tlabel(0.25,3.1){\scriptsize $1$}

\xmarks{-3 step 3 until 3}

\ymarks{-3 step 3 until 3}

\dotted \polyline{(0,0), (4.801,-1.397)}

\drawcolor[gray]{0.7}

\circle{(0,0),3}

\drawcolor[rgb]{0.33,0.33,0.33}

\penwd{1.5pt}

\arrow \parafcn{0, 343, 5}{3*dir(t)}

\end{mfpic}

\item $t = -2$

\begin{mfpic}[10]{-5}{5}{-5}{5}

\axes

\tlabel(4.5,-0.5){\scriptsize $x$}

\tlabel(0.5,4.5){\scriptsize $y$}

\tlabel(3.1,-0.75){\scriptsize $1$}

\tlabel(0.25,3.1){\scriptsize $1$}

\xmarks{-3 step 3 until 3}

\ymarks{-3 step 3 until 3}

\dotted \polyline{(0,0), (-2.081,-4.546)}

\drawcolor[gray]{0.7}

\circle{(0,0),3}

\drawcolor[rgb]{0.33,0.33,0.33}

\penwd{1.5pt}

\arrow \parafcn{0, 114, 5}{3*dir(-t)}

\end{mfpic}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

%\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $t = 12$ (between 1 and 2 revolutions)

\begin{mfpic}[10]{-5}{5}{-5}{5}

\axes

\tlabel(5,-0.5){\scriptsize $x$}

\tlabel(0.5,5){\scriptsize $y$}

\tlabel(3.1,-0.75){\scriptsize $1$}

\tlabel(0.25,3.1){\scriptsize $1$}

\xmarks{-3 step 3 until 3}

\ymarks{-3 step 3 until 3}

\dotted \polyline{(0,0), (4.219,-2.683)}

\drawcolor[gray]{0.7}

\circle{(0,0),3}

\drawcolor[rgb]{0.33,0.33,0.33}

\penwd{1.5pt}

\arrow \parafcn{0, 687, 5}{3*dir(t)}

\end{mfpic}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

%\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item About 30.12 miles per hour

\item About 6274.52 revolutions per minute

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item About 3.33 miles per hour

\item About 53.55 miles per hour

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item 70 miles per hour

\item About 4.32 miles per hour

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\addtocounter{enumi}{1}

\item $12\pi$ square units

\item $6250\pi$ square units

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $79.2825\pi \approx 249.07$ square units \vphantom{$\dfrac{\pi}{2}$}

\item $\dfrac{\pi}{2}$ square units

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\dfrac{50\pi}{3}$ square units

\item $38.025 \pi \approx 119.46$ square units \vphantom{$\dfrac{50\pi}{3}$}

\end{enumerate}

\end{multicols}

\closegraphsfile

10.2: The Unit Circle: Cosine and Sine

\subsection{Exercises}

In Exercises \ref{valuefirst} - \ref{valuelast}, find the exact value of the cosine and sine of the given angle.

\begin{multicols}{4}

\begin{enumerate}

\item $\theta = 0$ \vphantom{$\dfrac{\pi}{4}$} \label{valuefirst}

\item $\theta = \dfrac{\pi}{4}$

\item $\theta = \dfrac{\pi}{3}$

\item $\theta = \dfrac{\pi}{2}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{4}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\theta = \dfrac{2\pi}{3}$

\item $\theta = \dfrac{3\pi}{4}$

\item $\theta = \pi$ \vphantom{$\dfrac{7\pi}{4}$}

\item $\theta = \dfrac{7\pi}{6}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{4}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\theta = \dfrac{5\pi}{4}$

\item $\theta = \dfrac{4\pi}{3}$

\item $\theta = \dfrac{3\pi}{2}$

\item $\theta = \dfrac{5\pi}{3}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{4}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\theta = \dfrac{7\pi}{4}$

\item $\theta = \dfrac{23\pi}{6}$

\item $\theta = -\dfrac{13\pi}{2}$

\item $\theta = -\dfrac{43\pi}{6}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{4}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\theta = -\dfrac{3\pi}{4}$

\item $\theta = -\dfrac{\pi}{6}$ \vphantom{$\dfrac{7\pi}{4}$}

\item $\theta = \dfrac{10\pi}{3}$

\item $\theta = 117\pi$ \vphantom{$\dfrac{7\pi}{4}$} \label{valuelast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

In Exercises \ref{findthevaluefirst} - \ref{findthevaluelast}, use the results developed throughout the section to find the requested value.

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item If $\sin(\theta) = -\dfrac{7}{25}$ with $\theta$ in Quadrant IV, what is $\cos(\theta)$? \label{findthevaluefirst}

\item If $\cos(\theta) = \dfrac{4}{9}$ with $\theta$ in Quadrant I, what is $\sin(\theta)$?

\item If $\sin(\theta) = \dfrac{5}{13}$ with $\theta$ in Quadrant II, what is $\cos(\theta)$?

\item If $\cos(\theta) = -\dfrac{2}{11}$ with $\theta$ in Quadrant III, what is $\sin(\theta)$?

\item If $\sin(\theta) = -\dfrac{2}{3}$ with $\theta$ in Quadrant III, what is $\cos(\theta)$?

\item If $\cos(\theta) = \dfrac{28}{53}$ with $\theta$ in Quadrant IV, what is $\sin(\theta)$?

\item If $\sin(\theta) = \dfrac{2\sqrt{5}}{5}$ and $\dfrac{\pi}{2} < \theta < \pi$, what is $\cos(\theta)$?

\item If $\cos(\theta) = \dfrac{\sqrt{10}}{10}$ and $2\pi < \theta < \dfrac{5\pi}{2}$, what is $\sin(\theta)$?

\item If $\sin(\theta) = -0.42$ and $\pi < \theta < \dfrac{3\pi}{2}$, what is $\cos(\theta)$?

\item If $\cos(\theta) = -0.98$ and $\dfrac{\pi}{2} < \theta < \pi$, what is $\sin(\theta)$? \label{findthevaluelast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\pagebreak

In Exercises \ref{solveforanglefirst} - \ref{solveforanglelast}, find all of the angles which satisfy the given equation.

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sin(\theta) = \dfrac{1}{2}$ \vphantom{$\dfrac{2}{2}$} \label{solveforanglefirst}

\item $\cos(\theta) = -\dfrac{\sqrt{3}}{2}$

\item $\sin(\theta) = 0$ \vphantom{$\dfrac{2}{2}$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos(\theta) = \dfrac{\sqrt{2}}{2}$

\item $\sin(\theta) = \dfrac{\sqrt{3}}{2}$

\item $\cos(\theta) = -1$ \vphantom{$\dfrac{\sqrt{2}}{2}$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sin(\theta) = -1$ \vphantom{$\dfrac{\sqrt{2}}{2}$}

\item $\cos(\theta) = \dfrac{\sqrt{3}}{2}$

\item $\cos(\theta) = -1.001$ \vphantom{$\dfrac{\sqrt{2}}{2}$} \label{solveforanglelast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

In Exercises \ref{solvefortfirst} - \ref{solvefortlast}, solve the equation for $t$. (See the comments following Theorem \ref{cosinesinefunctiondomainrange}.)

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos(t) = 0$ \vphantom{$\dfrac{\sqrt{2}}{2}$} \label{solvefortfirst}

\item $\sin(t) = -\dfrac{\sqrt{2}}{2}$

\item $\cos(t) = 3$ \vphantom{$\dfrac{\sqrt{2}}{2}$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sin(t) = -\dfrac{1}{2}$

\item $\cos(t) = \dfrac{1}{2}$

\item $\sin(t) = -2$ \vphantom{$\dfrac{1}{2}$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos(t) = 1$ \vphantom{$\dfrac{\sqrt{2}}{2}$}

\item $\sin(t) = 1$ \vphantom{$\dfrac{\sqrt{2}}{2}$}

\item $\cos(t) = -\dfrac{\sqrt{2}}{2}$ \label{solvefortlast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

In Exercises \ref{calculatorfirst} - \ref{calculatorlast}, use your calculator to approximate the given value to three decimal places. Make sure your calculator is in the proper angle measurement mode!

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sin(78.95^{\circ})$ \label{calculatorfirst}

\item $\cos(-2.01)$

\item $\sin(392.994)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos(207^{\circ})$

\item $\sin\left( \pi^{\circ} \right)$

\item $\cos(e)$ \label{calculatorlast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

In Exercises \ref{firsttriangle} - \ref{lasttriangle}, find the measurement of the missing angle and the lengths of the missing sides. (See Example \ref{righttriangleex})

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item Find $\theta$, $b$, and $c$. \label{firsttriangle}

\begin{mfpic}[15]{-5}{5}{-5}{5}

\polyline{(-4.330,0), (4.330,0), (4.330,5), (-4.330,0)}

\arrow \reverse \arrow \shiftpath{(-4.330,0)} \parafcn{5, 25, 5}{3*dir(t)}

\arrow \reverse \arrow \shiftpath{(4.330,5)} \parafcn{215, 265, 5}{1.5*dir(t)}

\tlabel(-1, 0.6){$30^{\circ}$}

\tlabel(0,-0.75){$1$}

\tlabel(4.75,2.25){$b$}

\tlabel(-0.5,3){$c$}

\tlabel(3,3){$\theta$}

\polyline{(3.93, 0), (3.93, 0.4), (4.33, 0.4)}

\end{mfpic}

\item Find $\theta$, $a$, and $c$.

\begin{mfpic}[18]{-5}{5}{-5}{5}

\polyline{(-2.5, 0), (2.5,0), (-2.5,5), (-2.5,0)}

\arrow \reverse \arrow \shiftpath{(2.5,0)} \parafcn{140, 175, 5}{1.5*dir(t)}

\arrow \reverse \arrow \shiftpath{(-2.5,5)} \parafcn{275, 310, 5}{1.5*dir(t)}

\tlabel(-2, 2.75){$45^{\circ}$}

\tlabel(-0.5,-0.75){$3$}

\tlabel(-3.25,2.25){$a$}

\tlabel(0,3){$c$}

\tlabel(0.5,0.5){$\theta$}

\polyline{(-2.5, 0.4), (-2.1, 0.4), (-2.1, 0)}

\end{mfpic}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\pagebreak

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item Find $\alpha$, $a$, and $b$.

\begin{mfpic}[18]{-1}{5}{-1}{7}

\polyline{(0,0), (0,6.709), (4.357, 6.709), (0,0)}

\arrow \reverse \arrow \parafcn{60, 87, 5}{1.75*dir(t)}

\arrow \reverse \arrow \shiftpath{(4.357,6.709)} \parafcn{185, 232, 5}{1.5*dir(t)}

\tlabel(0.25, 2){$33^{\circ}$}

\tlabel(3,3){$8$}

\tlabel(2,7){$b$}

\tlabel(-0.75,4){$a$}

\tlabel(2.25,5.75){$\alpha$}

\polyline{(0,6.304), (0.4, 6.304), (0.4, 6.704)}

\end{mfpic}

\item Find $\beta$, $a$, and $c$. \label{lasttriangle}

\begin{mfpic}[18]{-6}{1}{-1}{9}

\polyline{(0,0), (0,6), (-5.402, 6), (0,0)}

\arrow \reverse \arrow \parafcn{95, 127, 5}{1.75*dir(t)}

\arrow \reverse \arrow \shiftpath{(-5.402,6)} \parafcn{317, 355, 5}{1.5*dir(t)}

\tlabel(-3.75, 5){$48^{\circ}$}

\tlabel(0.5,3){$6$}

\tlabel(-2.6,6.25){$a$}

\tlabel(-3.25,2.5){$c$}

\tlabel(-1,2){$\beta$}

\polyline{(0,5.6), (-0.4, 5.6), (-0.4, 6)}

\end{mfpic}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

In Exercises \ref{missingsidefirst} - \ref{missingsidelast}, assume that $\theta$ is an acute angle in a right triangle and use Theorem \ref{cosinesinetriangle} to find the requested side.

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item If $\theta = 12^{\circ}$ and the side adjacent to $\theta$ has length 4, how long is the hypotenuse? \label{missingsidefirst}

\item If $\theta = 78.123^{\circ}$ and the hypotenuse has length 5280, how long is the side adjacent to $\theta$?

\item If $\theta = 59^{\circ}$ and the side opposite $\theta$ has length 117.42, how long is the hypotenuse?

\item If $\theta = 5^{\circ}$ and the hypotenuse has length 10, how long is the side opposite $\theta$?

\item If $\theta = 5^{\circ}$ and the hypotenuse has length 10, how long is the side adjacent to $\theta$?

\item If $\theta = 37.5^{\circ}$ and the side opposite $\theta$ has length 306, how long is the side adjacent to $\theta$? \label{missingsidelast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

In Exercises \ref{pointsfirst} - \ref{pointslast}, let $\theta$ be the angle in standard position whose terminal side contains the given point then compute $\cos(\theta)$ and $\sin(\theta)$.

\begin{multicols}{4}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $P(-7, 24)$ \label{pointsfirst}

\item $Q(3, 4)$

\item $R(5, -9)$

\item $T(-2, -11)$ \label{pointslast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

In Exercises \ref{motionfirst} - \ref{motionlast}, find the equations of motion for the given scenario. Assume that the center of the motion is the origin, the motion is counter-clockwise and that $t = 0$ corresponds to a position along the positive $x$-axis. (See Equation \ref{equationsforcircularmotion} and Example \ref{EarthRotationEx}.)

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \label{motionfirst} A point on the edge of the spinning yo-yo in Exercise \ref{spinningyoyo} from Section \ref{Angles}.

Recall: The diameter of the yo-yo is 2.25 inches and it spins at 4500 revolutions per minute.

\item The yo-yo in exercise \ref{yoyotrick} from Section \ref{Angles}.

Recall: The radius of the circle is 28 inches and it completes one revolution in 3 seconds.

\item A point on the edge of the hard drive in Exercise \ref{harddrive} from Section \ref{Angles}.

Recall: The diameter of the hard disk is 2.5 inches and it spins at 7200 revolutions per minute.

\item \label{motionlast} A passenger on the Big Wheel in Exercise \ref{giantwheelmotion} from Section \ref{Angles}.

Recall: The diameter is 128 feet and completes 2 revolutions in 2 minutes, 7 seconds.

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item Consider the numbers: $0$, $1$, $2$, $3$, $4$. Take the square root of each of these numbers, then divide each by $2$. The resulting numbers should look hauntingly familiar. (See the values in the table on \pageref{CosineSineFacts}.)

\item Let $\alpha$ and $\beta$ be the two acute angles of a right triangle. (Thus $\alpha$ and $\beta$ are complementary angles.) Show that $\sin(\alpha) = \cos(\beta)$ and $\sin(\beta) = \cos(\alpha)$. The fact that co-functions of complementary angles are equal in this case is not an accident and a more general result will be given in Section \ref{Identities}.

\label{cofunctionforeshadowing}

\item In the scenario of Equation \ref{equationsforcircularmotion}, we assumed that at $t=0$, the object was at the point $(r,0)$. If this is not the case, we can adjust the equations of motion by introducing a `time delay.' If $t_{0} > 0$ is the first time the object passes through the point $(r,0)$, show, with the help of your classmates, the equations of motion are $x = r \cos(\omega (t - t_{0}))$ and $y = r \sin(\omega (t-t_{0}))$.

\end{enumerate}

\newpage

\subsection{Answers}

\begin{multicols}{2}

\begin{enumerate}

\item $\cos(0) = 1$, $\; \sin(0) = 0$ \vphantom{$\dfrac{\sqrt{2}}{2}$}

\item $\cos \left(\dfrac{\pi}{4} \right) = \dfrac{\sqrt{2}}{2}$, $\; \sin \left(\dfrac{\pi}{4} \right) = \dfrac{\sqrt{2}}{2}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos \left(\dfrac{\pi}{3}\right) = \dfrac{1}{2}$, $\; \sin \left(\dfrac{\pi}{3}\right) = \dfrac{\sqrt{3}}{2}$

\item $\cos \left(\dfrac{\pi}{2}\right) = 0$, $\; \sin \left(\dfrac{\pi}{2}\right) = 1$ \vphantom{$\dfrac{\sqrt{2}}{2}$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos\left(\dfrac{2\pi}{3}\right) = -\dfrac{1}{2}$, $\; \sin \left(\dfrac{2\pi}{3}\right) = \dfrac{\sqrt{3}}{2}$

\item $\cos \left(\dfrac{3\pi}{4} \right) = -\dfrac{\sqrt{2}}{2}$, $\; \sin \left(\dfrac{3\pi}{4} \right) = \dfrac{\sqrt{2}}{2}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos(\pi) = -1$, $\; \sin(\pi) = 0$ \vphantom{$\dfrac{\sqrt{3}}{2}$}

\item $\cos\left(\dfrac{7\pi}{6}\right) = -\dfrac{\sqrt{3}}{2}$, $\; \sin\left(\dfrac{7\pi}{6}\right) = -\dfrac{1}{2}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos \left(\dfrac{5\pi}{4} \right) = -\dfrac{\sqrt{2}}{2}$, $\; \sin \left(\dfrac{5\pi}{4} \right) = -\dfrac{\sqrt{2}}{2}$

\item $\cos\left(\dfrac{4\pi}{3}\right) = -\dfrac{1}{2}$, $\; \sin \left(\dfrac{4\pi}{3}\right) = -\dfrac{\sqrt{3}}{2}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos \left(\dfrac{3\pi}{2}\right) = 0$, $\; \sin \left(\dfrac{3\pi}{2}\right) = -1$

\item $\cos\left(\dfrac{5\pi}{3}\right) = \dfrac{1}{2}$, $\; \sin \left(\dfrac{5\pi}{3}\right) = -\dfrac{\sqrt{3}}{2}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos \left(\dfrac{7\pi}{4} \right) = \dfrac{\sqrt{2}}{2}$, $\; \sin \left(\dfrac{7\pi}{4} \right) = -\dfrac{\sqrt{2}}{2}$

\item $\cos\left(\dfrac{23\pi}{6}\right) = \dfrac{\sqrt{3}}{2}$, $\; \sin\left(\dfrac{23\pi}{6}\right) = -\dfrac{1}{2}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos \left(-\dfrac{13\pi}{2}\right) = 0$, $\; \sin \left(-\dfrac{13\pi}{2}\right) = -1$ \vphantom{$\dfrac{\sqrt{3}}{2}$}

\item $\cos\left(-\dfrac{43\pi}{6}\right) = -\dfrac{\sqrt{3}}{2}$, $\; \sin\left(-\dfrac{43\pi}{6}\right) = \dfrac{1}{2}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos \left(-\dfrac{3\pi}{4} \right) = -\dfrac{\sqrt{2}}{2}$, $\; \sin \left(-\dfrac{3\pi}{4} \right) = -\dfrac{\sqrt{2}}{2}$

\item $\cos\left(-\dfrac{\pi}{6}\right) = \dfrac{\sqrt{3}}{2}$, $\; \sin\left(-\dfrac{\pi}{6}\right) = -\dfrac{1}{2}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos\left(\dfrac{10\pi}{3}\right) = -\dfrac{1}{2}$, $\; \sin \left(\dfrac{10\pi}{3}\right) = -\dfrac{\sqrt{3}}{2}$

\item $\cos(117\pi) = -1$, $\; \sin(117\pi) = 0$ \vphantom{$\dfrac{\sqrt{3}}{2}$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item If $\sin(\theta) = -\dfrac{7}{25}$ with $\theta$ in Quadrant IV, then $\cos(\theta) = \dfrac{24}{25}$.

\item If $\cos(\theta) = \dfrac{4}{9}$ with $\theta$ in Quadrant I, then $\sin(\theta) = \dfrac{\sqrt{65}}{9}$.

\item If $\sin(\theta) = \dfrac{5}{13}$ with $\theta$ in Quadrant II, then $\cos(\theta) = -\dfrac{12}{13}$.

\item If $\cos(\theta) = -\dfrac{2}{11}$ with $\theta$ in Quadrant III, then $\sin(\theta) = -\dfrac{\sqrt{117}}{11}$.

\item If $\sin(\theta) = -\dfrac{2}{3}$ with $\theta$ in Quadrant III, then $\cos(\theta) = -\dfrac{\sqrt{5}}{3}$.

\item If $\cos(\theta) = \dfrac{28}{53}$ with $\theta$ in Quadrant IV, then $\sin(\theta) = -\dfrac{45}{53}$.

\item If $\sin(\theta) = \dfrac{2\sqrt{5}}{5}$ and $\dfrac{\pi}{2} < \theta < \pi$, then $\cos(\theta) = -\dfrac{\sqrt{5}}{5}$.

\item If $\cos(\theta) = \dfrac{\sqrt{10}}{10}$ and $2\pi < \theta < \dfrac{5\pi}{2}$, then $\sin(\theta) = \dfrac{3 \sqrt{10}}{10}$.

\item If $\sin(\theta) = -0.42$ and $\pi < \theta < \dfrac{3\pi}{2}$, then $\cos(\theta) = -\sqrt{0.8236} \approx -0.9075$.

\item If $\cos(\theta) = -0.98$ and $\dfrac{\pi}{2} < \theta < \pi$, then $\sin(\theta) = \sqrt{0.0396} \approx 0.1990$.

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sin(\theta) = \dfrac{1}{2}$ when $\theta = \dfrac{\pi}{6} + 2\pi k$ or $\theta = \dfrac{5\pi}{6} + 2\pi k$ for any integer $k$.

\item $\cos(\theta) = -\dfrac{\sqrt{3}}{2}$ when $\theta = \dfrac{5\pi}{6} + 2\pi k$ or $\theta = \dfrac{7\pi}{6} + 2\pi k$ for any integer $k$.

\item $\sin(\theta) = 0$ when $\theta = \pi k$ for any integer $k$.

\item $\cos(\theta) = \dfrac{\sqrt{2}}{2}$ when $\theta = \dfrac{\pi}{4} + 2\pi k$ or $\theta = \dfrac{7\pi}{4} + 2\pi k$ for any integer $k$.

\item $\sin(\theta) = \dfrac{\sqrt{3}}{2}$ when $\theta = \dfrac{\pi}{3} + 2\pi k$ or $\theta = \dfrac{2\pi}{3} + 2\pi k$ for any integer $k$.

\item $\cos(\theta) = -1$ when $\theta = (2k + 1)\pi$ for any integer $k$.

\item $\sin(\theta) = -1$ when $\theta = \dfrac{3\pi}{2} + 2\pi k$ for any integer $k$.

\item $\cos(\theta) = \dfrac{\sqrt{3}}{2}$ when $\theta = \dfrac{\pi}{6} + 2\pi k$ or $\theta = \dfrac{11\pi}{6} + 2\pi k$ for any integer $k$.

%\item $\sin(\theta) = \dfrac{\sqrt{2}}{2}$ when $\theta = \dfrac{\pi}{4} + 2\pi k$ or $\theta = \dfrac{3\pi}{4} + 2\pi k$ for any integer $k$.

\item $\cos(\theta) = -1.001$ never happens

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos(t) = 0$ when $t = \dfrac{\pi}{2} + \pi k$ for any integer $k$.

\item $\sin(t) = -\dfrac{\sqrt{2}}{2}$ when $t = \dfrac{5\pi}{4} + 2\pi k$ or $t = \dfrac{7\pi}{4} + 2\pi k$ for any integer $k$.

\item $\cos(t) = 3$ never happens.

\item $\sin(t) = -\dfrac{1}{2}$ when $t = \dfrac{7\pi}{6} + 2\pi k$ or $t = \dfrac{11\pi}{6} + 2\pi k$ for any integer $k$.

\item $\cos(t) = \dfrac{1}{2}$ when $t = \dfrac{\pi}{3} + 2\pi k$ or $t = \dfrac{5\pi}{3} + 2\pi k$ for any integer $k$.

\item $\sin(t) = -2$ never happens

\item $\cos(t) = 1$ when $t = 2\pi k$ for any integer $k$.

\item $\sin(t) = 1$ when $t = \dfrac{\pi}{2} + 2\pi k$ for any integer $k$.

\item $\cos(t) = -\dfrac{\sqrt{2}}{2}$ when $t = \dfrac{3\pi}{4} + 2\pi k$ or $t = \dfrac{5\pi}{4} + 2\pi k$ for any integer $k$.

%\item $\sin(t) = -\dfrac{\sqrt{3}}{2}$ when $t = \dfrac{4\pi}{3} + 2\pi k$ or $t = \dfrac{5\pi}{3} + 2\pi k$ for any integer $k$.

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sin(78.95^{\circ}) \approx 0.981$

\item $\cos(-2.01) \approx -0.425$

\item $\sin(392.994) \approx -0.291$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos(207^{\circ}) \approx -0.891$

\item $\sin\left( \pi^{\circ} \right) \approx 0.055$

\item $\cos(e) \approx -0.912$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\theta = 60^{\circ}$, $b = \dfrac{ \sqrt{3}}{3}$, $c=\dfrac{2 \sqrt{3}}{3}$

\item $\theta = 45^{\circ}$, $a = 3$, $c = 3\sqrt{2}$

\item $\alpha = 57^{\circ}$, $a = 8 \cos(33^{\circ}) \approx 6.709$, $b = 8 \sin(33^{\circ}) \approx 4.357$

\item $\beta = 42^{\circ}$, $c = \dfrac{6}{\sin(48^{\circ})} \approx 8.074$, $a = \sqrt{c^2 - 6^2} \approx 5.402$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item The hypotenuse has length $\dfrac{4}{\cos(12^{\circ})}\approx 4.089$.

\item The side adjacent to $\theta$ has length $5280\cos(78.123^{\circ}) \approx 1086.68$.

\item The hypotenuse has length $\dfrac{117.42}{\sin(59^{\circ})}\approx 136.99$.

\item The side opposite $\theta$ has length $10\sin(5^{\circ}) \approx 0.872$.

\item The side adjacent to $\theta$ has length $10\cos(5^{\circ}) \approx 9.962$.

\item The hypotenuse has length $c = \dfrac{306}{\sin(37.5^{\circ})}\approx 502.660$, so the side adjacent to $\theta$ has length $\sqrt{c^2 - 306^{2}} \approx 398.797$.

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos(\theta) = -\dfrac{7}{25}, \; \sin(\theta) = \dfrac{24}{25}$

\item $\cos(\theta) = \dfrac{3}{5}, \; \sin(\theta) = \dfrac{4}{5}$

\item $\cos(\theta) = \dfrac{5\sqrt{106}}{106}, \; \sin(\theta) = -\dfrac{9\sqrt{106}}{106}$

\item $\cos(\theta) = -\dfrac{2\sqrt{5}}{25}, \; \sin(\theta) = -\dfrac{11\sqrt{5}}{25}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $r = 1.125$ inches, $\omega = 9000 \pi \, \frac{\text{radians}}{\text{minute}}$, $x = 1.125 \cos(9000 \pi \, t)$, $y = 1.125 \sin(9000 \pi \, t)$. Here $x$ and $y$ are measured in inches and $t$ is measured in minutes.

\item $r = 28$ inches, $\omega = \frac{2\pi}{3} \, \frac{\text{radians}}{\text{second}}$, $x = 28 \cos\left(\frac{2\pi}{3} \, t \right)$, $y = 28 \sin\left(\frac{2\pi}{3} \, t \right)$. Here $x$ and $y$ are measured in inches and $t$ is measured in seconds.

\item $r = 1.25$ inches, $\omega = 14400 \pi \, \frac{\text{radians}}{\text{minute}}$, $x = 1.25 \cos(14400 \pi \, t)$, $y = 1.25 \sin(14400 \pi \, t)$. Here $x$ and $y$ are measured in inches and $t$ is measured in minutes.

\item $r = 64$ feet, $\omega = \frac{4\pi}{127} \, \frac{\text{radians}}{\text{second}}$, $x = 64 \cos\left(\frac{4\pi}{127} \, t \right)$, $y = 64 \sin\left(\frac{4\pi}{127} \, t \right)$. Here $x$ and $y$ are measured in feet and $t$ is measured in seconds

\end{enumerate}

\closegraphsfile

10.3: The Six Circular Functions and Fundamental Identities

\subsection{Exercises}

In Exercises \ref{circvaluefirst} - \ref{circvaluelast}, find the exact value or state that it is undefined.

\begin{multicols}{4}

\begin{enumerate}

\item $\tan \left( \dfrac{\pi}{4} \right)$ \vphantom{$\csc \left( \dfrac{5\pi}{6} \right)$} \label{circvaluefirst}

\item $\sec \left( \dfrac{\pi}{6} \right)$ \vphantom{$\csc \left( \dfrac{5\pi}{6} \right)$}

\item $\csc \left( \dfrac{5\pi}{6} \right)$

\item $\cot \left( \dfrac{4\pi}{3} \right)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{4}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\tan \left( -\dfrac{11\pi}{6} \right)$

\item $\sec \left( -\dfrac{3\pi}{2} \right)$

\item $\csc \left( -\dfrac{\pi}{3} \right)$ \vphantom{$\csc \left( \dfrac{5\pi}{6} \right)$}

\item $\cot \left( \dfrac{13\pi}{2} \right)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{4}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\tan \left( 117\pi \right)$ \vphantom{$\csc \left( \dfrac{5\pi}{6} \right)$}

\item $\sec \left( -\dfrac{5\pi}{3} \right)$

\item $\csc \left( 3\pi \right)$ \vphantom{$\csc \left( \dfrac{5\pi}{6} \right)$}

\item $\cot \left( -5\pi \right)$ \vphantom{$\csc \left( \dfrac{5\pi}{6} \right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{4}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\tan \left( \dfrac{31\pi}{2} \right)$

\item $\sec \left( \dfrac{\pi}{4} \right)$ \vphantom{$\csc \left( \dfrac{5\pi}{6} \right)$}

\item $\csc \left( -\dfrac{7\pi}{4} \right)$

\item $\cot \left( \dfrac{7\pi}{6} \right)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{4}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\tan \left( \dfrac{2\pi}{3} \right)$

\item $\sec \left( -7\pi \right)$ \vphantom{$\csc \left( \dfrac{5\pi}{6} \right)$}

\item $\csc \left( \dfrac{\pi}{2} \right)$ \vphantom{$\csc \left( \dfrac{5\pi}{6} \right)$}

\item $\cot \left( \dfrac{3\pi}{4} \right)$ \label{circvaluelast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

In Exercises \ref{findothercircfirst} - \ref{findothercirclast}, use the given the information to find the exact values of the remaining circular functions of $\theta$.

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sin(\theta) = \dfrac{3}{5}$ with $\theta$ in Quadrant II \label{findothercircfirst}

\item $\tan(\theta) = \dfrac{12}{5}$ with $\theta$ in Quadrant III

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\csc(\theta) = \dfrac{25}{24}$ with $\theta$ in Quadrant I

\item $\sec(\theta) = 7$ with $\theta$ in Quadrant IV \vphantom{$\dfrac{25}{24}$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\csc(\theta) = -\dfrac{10\sqrt{91}}{91}$ with $\theta$ in Quadrant III

\item $\cot(\theta) = -23$ with $\theta$ in Quadrant II \vphantom{$\dfrac{10}{\sqrt{91}}$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\tan(\theta) = -2$ with $\theta$ in Quadrant IV.

\item $\sec(\theta) = -4$ with $\theta$ in Quadrant II.

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cot(\theta) = \sqrt{5}$ with $\theta$ in Quadrant III. \vphantom{$\dfrac{25}{24}$}

\item $\cos(\theta) = \dfrac{1}{3}$ with $\theta$ in Quadrant I.

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cot(\theta) = 2$ with $0 < \theta < \dfrac{\pi}{2}$.

\item $\csc(\theta) = 5$ with $\dfrac{\pi}{2} < \theta < \pi$.

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\tan(\theta) = \sqrt{10}$ with $\pi < \theta < \dfrac{3\pi}{2}$.

\item $\sec(\theta) = 2\sqrt{5}$ with $\dfrac{3\pi}{2} < \theta < 2\pi$. \label{findothercirclast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

In Exercises \ref{circcalcfirst} - \ref{circcalclast}, use your calculator to approximate the given value to three decimal places. Make sure your calculator is in the proper angle measurement mode!

\begin{multicols}{4}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\csc(78.95^{\circ})$ \label{circcalcfirst}

\item $\tan(-2.01)$

\item $\cot(392.994)$

\item $\sec(207^{\circ})$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{4}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\csc(5.902)$

\item $\tan(39.672^{\circ})$

\item $\cot(3^{\circ})$

\item $\sec(0.45)$ \label{circcalclast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\pagebreak

In Exercises \ref{circequanglefirst} - \ref{circequanglelast}, find all of the angles which satisfy the equation.

\begin{multicols}{4}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\tan(\theta) = \sqrt{3}$ \vphantom{$\dfrac{\sqrt{3}}{3}$} \label{circequanglefirst}

\item $\sec(\theta) = 2$ \vphantom{$\dfrac{\sqrt{3}}{3}$}

\item $\csc(\theta) = -1$ \vphantom{$\dfrac{\sqrt{3}}{3}$}

\item $\cot(\theta) = \dfrac{\sqrt{3}}{3}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{4}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\tan(\theta) = 0$

\item $\sec(\theta) = 1$

\item $\csc(\theta) = 2$

\item $\cot(\theta) = 0$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{4}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\tan(\theta) = -1$ \vphantom{$\dfrac{1}{2}$}

\item $\sec(\theta) = 0$ \vphantom{$\dfrac{1}{2}$}

\item $\csc(\theta) = -\dfrac{1}{2}$

\item $\sec(\theta) = -1$ \vphantom{$\dfrac{1}{2}$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{4}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\tan(\theta) = -\sqrt{3}$

\item $\csc(\theta) = -2$ \vphantom{$\sqrt{3}$}

\item $\cot(\theta) = -1$ \vphantom{$\sqrt{3}$} \label{circequanglelast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

In Exercises \ref{circequtfirst} - \ref{circequtlast}, solve the equation for $t$. Give exact values.

\begin{multicols}{4}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cot(t) = 1$ \vphantom{$\dfrac{2\sqrt{3}}{3}$} \label{circequtfirst}

\item $\tan(t) = \dfrac{\sqrt{3}}{3}$ \vphantom{$\dfrac{2\sqrt{3}}{3}$}

\item $\sec(t) = -\dfrac{2\sqrt{3}}{3}$

\item $\csc(t) = 0$ \vphantom{$\dfrac{2\sqrt{3}}{3}$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{4}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cot(t) = -\sqrt{3}$ \vphantom{$\dfrac{2\sqrt{3}}{3}$}

\item $\tan(t) = -\dfrac{\sqrt{3}}{3}$

\item $\sec(t) = \dfrac{2\sqrt{3}}{3}$

\item $\csc(t) = \dfrac{2\sqrt{3}}{3}$ \label{circequtlast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

In Exercises \ref{trianglecircfirst} - \ref{trianglecirclast}, use Theorem \ref{circularfunctionstriangle} to find the requested quantities.

\begin{multicols}{2} \raggedcolumns

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item Find $\theta$, $a$, and $c$. \label{trianglecircfirst}

\begin{mfpic}[15]{-5}{5}{-5}{5}

\polyline{(-4.330,0), (4.330,0), (4.330,5), (-4.330,0)}

\arrow \reverse \arrow \shiftpath{(-4.330,0)} \parafcn{5, 25, 5}{3*dir(t)}

\arrow \reverse \arrow \shiftpath{(4.330,5)} \parafcn{215, 265, 5}{1.5*dir(t)}

\tlabel(-1.25, 0.6){$\theta$}

\tlabel(0,-0.75){$9$}

\tlabel(4.75,2.25){$a$}

\tlabel(-0.5,3){$c$}

\tlabel(2.75,2.85){$60^{\circ}$}

\polyline{(3.93, 0), (3.93, 0.4), (4.33, 0.4)}

\end{mfpic}

\vspace{.5in}

\item Find $\alpha$, $b$, and $c$.

\begin{mfpic}[15]{-1}{5}{-1}{7}

\polyline{(0,0), (0,6.709), (4.357, 6.709), (0,0)}

\arrow \reverse \arrow \parafcn{60, 87, 5}{1.75*dir(t)}

\arrow \reverse \arrow \shiftpath{(4.357,6.709)} \parafcn{185, 232, 5}{1.5*dir(t)}

\tlabel(0.25, 2){$34^{\circ}$}

\tlabel(2.5,3){$c$}

\tlabel(2,7){$b$}

\tlabel(-0.85,4){$12$}

\tlabel(2.25,5.75){$\alpha$}

\polyline{(0,6.304), (0.4, 6.304), (0.4, 6.704)}

\end{mfpic}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\enlargethispage{.3in}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item Find $\theta$, $a$, and $c$.

\begin{mfpic}[18]{-5}{5}{-5}{5}

\polyline{(-2.5, 0), (2.5,0), (-2.5,5), (-2.5,0)}

\arrow \reverse \arrow \shiftpath{(2.5,0)} \parafcn{140, 175, 5}{1.5*dir(t)}

\arrow \reverse \arrow \shiftpath{(-2.5,5)} \parafcn{275, 310, 5}{1.5*dir(t)}

\tlabel(-2, 2.75){$47^{\circ}$}

\tlabel(-0.5,-0.75){$6$}

\tlabel(-3.25,2.25){$a$}

\tlabel(0,3){$c$}

\tlabel(0.5,0.5){$\theta$}

\polyline{(-2.5, 0.4), (-2.1, 0.4), (-2.1, 0)}

\end{mfpic}

\item Find $\beta$, $b$, and $c$. \label{trianglecirclast}

\begin{mfpic}[18]{-6}{1}{-1}{9}

\polyline{(0,0), (0,6), (-5.402, 6), (0,0)}

\arrow \reverse \arrow \parafcn{95, 127, 5}{1.75*dir(t)}

\arrow \reverse \arrow \shiftpath{(-5.402,6)} \parafcn{317, 355, 5}{1.5*dir(t)}

\tlabel(-3.75, 5){$\beta$}

\tlabel(0.5,3){$2.5$}

\tlabel(-2.6,6.25){$b$}

\tlabel(-3.25,2.5){$c$}

\tlabel(-1.2,2){$50^{\circ}$}

\polyline{(0,5.6), (-0.4, 5.6), (-0.4, 6)}

\end{mfpic}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

In Exercises \ref{moretrianglecircfirst} - \ref{moretrianglecirclast}, use Theorem \ref{circularfunctionstriangle} to answer the question. Assume that $\theta$ is an angle in a right triangle.

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item If $\theta = 30^{\circ}$ and the side opposite $\theta$ has length $4$, how long is the side adjacent to $\theta$? \label{moretrianglecircfirst}

\item If $\theta = 15^{\circ}$ and the hypotenuse has length $10$, how long is the side opposite $\theta$?

\item If $\theta = 87^{\circ}$ and the side adjacent to $\theta$ has length $2$, how long is the side opposite $\theta$?

\item If $\theta = 38.2^{\circ}$ and the side opposite $\theta$ has lengh $14$, how long is the hypoteneuse?

\item If $\theta = 2.05^{\circ}$ and the hypotenuse has length $3.98$, how long is the side adjacent to $\theta$?

\item If $\theta = 42^{\circ}$ and the side adjacent to $\theta$ has length $31$, how long is the side opposite $\theta$? \label{moretrianglecirclast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item A tree standing vertically on level ground casts a 120 foot long shadow. The angle of elevation from the end of the shadow to the top of the tree is $21.4^{\circ}$. Find the height of the tree to the nearest foot. With the help of your classmates, research the term \emph{umbra versa} and see what it has to do with the shadow in this problem.

\item The broadcast tower for radio station WSAZ (Home of ``Algebra in the Morning with Carl and Jeff'') has two enormous flashing red lights on it: one at the very top and one a few feet below the top. From a point 5000 feet away from the base of the tower on level ground the angle of elevation to the top light is $7.970^{\circ}$ and to the second light is $7.125^{\circ}$. Find the distance between the lights to the nearest foot.

\item On page \pageref{angleofelevation} we defined the angle of inclination (also known as the angle of elevation) and in this exercise we introduce a related angle - \index{angle ! of depression} the angle of depression (also known as \index{angle ! of declination} the angle of declination). The angle of depression of an object refers to the angle whose initial side is a horizontal line above the object and whose terminal side is the line-of-sight to the object below the horizontal. This is represented schematically below.

\label{angleofdepression}

\begin{center}

\begin{mfpic}[18]{-5}{5}{-5}{5}

\polyline{(-5,5), (4.330,5)}

\point[3pt]{(4.330,5)}

\dashed \polyline{(-4.330,0), (4.330,5)}

\reverse \arrow \shiftpath{(4.330,5)} \parafcn{185, 205, 5}{3*dir(t)}

\tlabel(0.75, 4){$\theta$}

\tlabel[cc](-1,5.5){horizontal}

\tlabel[cc](5.25,4.75){observer}

\plotsymbol[3pt]{Asterisk}{(-4.330,0)}

\tlabel(-5.0,-0.75){object}

\end{mfpic}

\smallskip

The angle of depression from the horizontal to the object is $\theta$

\end{center}

\begin{enumerate}

\item Show that if the horizontal is above and parallel to level ground then the angle of depression (from observer to object) and the angle of inclination (from object to observer) will be congruent because they are alternate interior angles.

\item \label{sasquatchfire} From a firetower 200 feet above level ground in the Sasquatch National Forest, a ranger spots a fire off in the distance. The angle of depression to the fire is $2.5^{\circ}$. How far away from the base of the tower is the fire?

\item The ranger in part \ref{sasquatchfire} sees a Sasquatch running directly from the fire towards the firetower. The ranger takes two sightings. At the first sighthing, the angle of depression from the tower to the Sasquatch is $6^{\circ}$. The second sighting, taken just 10 seconds later, gives the the angle of depression as $6.5^{\circ}$. How far did the Saquatch travel in those 10 seconds? Round your answer to the nearest foot. How fast is it running in miles per hour? Round your answer to the nearest mile per hour. If the Sasquatch keeps up this pace, how long will it take for the Sasquatch to reach the firetower from his location at the second sighting? Round your answer to the nearest minute.

\end{enumerate}

\item When I stand 30 feet away from a tree at home, the angle of elevation to the top of the tree is $50^{\circ}$ and the angle of depression to the base of the tree is $10^{\circ}$. What is the height of the tree? Round your answer to the nearest foot.

\item From the observation deck of the lighthouse at Sasquatch Point 50 feet above the surface of Lake Ippizuti, a lifeguard spots a boat out on the lake sailing directly toward the lighthouse. The first sighting had an angle of depression of $8.2^{\circ}$ and the second sighting had an angle of depression of $25.9^{\circ}$. How far had the boat traveled between the sightings?

\item A guy wire 1000 feet long is attached to the top of a tower. When pulled taut it makes a $43^{\circ}$ angle with the ground. How tall is the tower? How far away from the base of the tower does the wire hit the ground?

\setcounter{HW}{\value{enumi}}

\end{enumerate}

In Exercises \ref{firstcirciden} - \ref{lastcirciden}, verify the identity. Assume that all quantities are defined.

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos(\theta) \sec(\theta) = 1$ \label{firstcirciden}

\item $\tan(\theta)\cos(\theta) = \sin(\theta)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sin(\theta) \csc(\theta) = 1$

\item $\tan(\theta) \cot(\theta) = 1$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\csc(\theta) \cos(\theta) = \cot(\theta)$ \vphantom{$\dfrac{\sin(\theta)}{\cos^{2}(\theta)}$}

\item $\dfrac{\sin(\theta)}{\cos^{2}(\theta)} = \sec(\theta) \tan(\theta)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\dfrac{\cos(\theta)}{\sin^{2}(\theta)} = \csc(\theta) \cot(\theta)$

\item $\dfrac{1+ \sin(\theta)}{\cos(\theta)} = \sec(\theta) + \tan(\theta)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\dfrac{1 - \cos(\theta)}{\sin(\theta)} = \csc(\theta) - \cot(\theta)$

\item $\dfrac{\cos(\theta)}{1 - \sin^{2}(\theta)} = \sec(\theta)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\dfrac{\sin(\theta)}{1 - \cos^{2}(\theta)} = \csc(\theta)$

\item $\dfrac{\sec(\theta)}{1 + \tan^{2}(\theta)} = \cos(\theta)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\dfrac{\csc(\theta)}{1 + \cot^{2}(\theta)} = \sin(\theta)$

\item $\dfrac{\tan(\theta)}{\sec^{2}(\theta) - 1} = \cot(\theta)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\dfrac{\cot(\theta)}{\csc^{2}(\theta) - 1} = \tan(\theta)$

\item $4 \cos^{2}(\theta) + 4 \sin^{2}(\theta) = 4$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $9 - \cos^{2}(\theta) - \sin^{2}(\theta) = 8$

\item $\tan^{3}(\theta) = \tan(\theta)\sec^{2}(\theta) - \tan(\theta)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sin^{5}(\theta) = \left(1-\cos^{2}(\theta)\right)^{2} \sin(\theta)$

\item $\sec^{10}(\theta) = \left(1 + \tan^{2}(\theta)\right)^4 \sec^{2}(\theta)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos^{2}(\theta)\tan^{3}(\theta) = \tan(\theta) - \sin(\theta)\cos(\theta)$

\item $\sec^{4}(\theta) - \sec^{2}(\theta) = \tan^{2}(\theta) + \tan^{4}(\theta)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\dfrac{\cos(\theta) + 1}{\cos(\theta) - 1} = \dfrac{1 + \sec(\theta)}{1 - \sec(\theta)}$

\item $\dfrac{\sin(\theta) + 1}{\sin(\theta) - 1} = \dfrac{1 + \csc(\theta)}{1 - \csc(\theta)}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\dfrac{1 - \cot(\theta)}{1+ \cot(\theta)} = \dfrac{\tan(\theta) - 1}{\tan(\theta) + 1}$

\item $\dfrac{1 - \tan(\theta)}{1+ \tan(\theta)} = \dfrac{\cos(\theta) - \sin(\theta)}{\cos(\theta) + \sin(\theta)}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\tan(\theta) + \cot(\theta) = \sec(\theta)\csc(\theta)$

\item $\csc(\theta) - \sin(\theta) = \cot(\theta)\cos(\theta)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos(\theta) - \sec(\theta) = -\tan(\theta)\sin(\theta)$

\item $\cos(\theta)(\tan(\theta) + \cot(\theta)) = \csc(\theta)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sin(\theta)(\tan(\theta) + \cot(\theta)) = \sec(\theta)$ \vphantom{$\dfrac{1}{1-\cos(\theta)}$}

\item $\dfrac{1}{1-\cos(\theta)} + \dfrac{1}{1+\cos(\theta)} = 2\csc^{2}(\theta)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\dfrac{1}{\sec(\theta) + 1} + \dfrac{1}{\sec(\theta)-1} = 2 \csc(\theta) \cot(\theta)$

\item $\dfrac{1}{\csc(\theta) + 1} + \dfrac{1}{\csc(\theta)-1} = 2 \sec(\theta) \tan(\theta)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\small

\item $\dfrac{1}{\csc(\theta)-\cot(\theta)} - \dfrac{1}{\csc(\theta) + \cot(\theta)} = 2 \cot(\theta)$

\item $\dfrac{\cos(\theta)}{1 - \tan(\theta)} + \dfrac{\sin(\theta)}{1 - \cot(\theta)} = \sin(\theta) + \cos(\theta)$

\normalsize

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\dfrac{1}{\sec(\theta) + \tan(\theta)} = \sec(\theta) - \tan(\theta)$

\item $\dfrac{1}{\sec(\theta) - \tan(\theta)} = \sec(\theta) + \tan(\theta)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\dfrac{1}{\csc(\theta) - \cot(\theta)} = \csc(\theta) + \cot(\theta)$

\item $\dfrac{1}{\csc(\theta) + \cot(\theta)} = \csc(\theta) - \cot(\theta)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\dfrac{1}{1-\sin(\theta)} = \sec^{2}(\theta) + \sec(\theta) \tan(\theta)$

\item $\dfrac{1}{1+\sin(\theta)} = \sec^{2}(\theta) - \sec(\theta) \tan(\theta)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\dfrac{1}{1-\cos(\theta)} = \csc^{2}(\theta) + \csc(\theta) \cot(\theta)$

\item $\dfrac{1}{1+\cos(\theta)} = \csc^{2}(\theta) - \csc(\theta) \cot(\theta)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\dfrac{\cos(\theta)}{1 + \sin(\theta)} = \dfrac{1-\sin(\theta)}{\cos(\theta)}$

\item $\csc(\theta) - \cot(\theta) = \dfrac{\sin(\theta)}{1 + \cos(\theta)}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\dfrac{1 - \sin(\theta)}{1 + \sin(\theta)} = (\sec(\theta) - \tan(\theta))^{2}$ \label{lastcirciden}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\pagebreak

In Exercises \ref{logcircidenfirst} - \ref{logcircidenlast}, verify the identity. You may need to consult Sections \ref{AbsoluteValueFunctions} and \ref{LogProperties} for a review of the properties of absolute value and logarithms before proceeding.

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\quad \ln|\sec(\theta)| = -\ln|\cos(\theta)|$ \label{logcircidenfirst}

\item $-\ln|\csc(\theta)| = \ln|\sin(\theta)|$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $-\ln|\sec(\theta) - \tan(\theta)| = \ln|\sec(\theta)+\tan(\theta)|$

\item $-\ln|\csc(\theta) + \cot(\theta)|= \ln|\csc(\theta) - \cot(\theta)|$ \label{logcircidenlast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item Verify the domains and ranges of the tangent, cosecant and cotangent functions as presented in Theorem \ref{circularfunctionsdomainrange}.

\item As we did in Exercise \ref{cofunctionforeshadowing} in Section \ref{TheUnitCircle}, let $\alpha$ and $\beta$ be the two acute angles of a right triangle. (Thus $\alpha$ and $\beta$ are complementary angles.) Show that $\sec(\alpha) = \csc(\beta)$ and $\tan(\alpha) = \cot(\beta)$. The fact that co-functions of complementary angles are equal in this case is not an accident and a more general result will be given in Section \ref{Identities}.

\label{cofunctionforeshadowingagain}

\item We wish to establish the inequality $\cos(\theta) < \dfrac{\sin(\theta)}{\theta} < 1$ for $0 < \theta < \dfrac{\pi}{2}.$ Use the diagram from the beginning of the section, partially reproduced below, to answer the following.

\begin{center}

\begin{mfpic}[20]{-1}{4}{-1}{6}

\axes

\drawcolor[gray]{0.7}

\parafcn{0,90,5}{3*dir(t)}

\drawcolor[rgb]{0.33,0.33,0.33}

\arrow \parafcn{5, 55, 5}{0.75*dir(t)}

\tlabel[cc](0.75,0.5){\scriptsize $\theta$}

\point[3pt]{(0,0), (3,5.196), (3,0)}

\tlabel(3.75,-0.5){\scriptsize $x$}

\tlabel(0.25,5.75){\scriptsize $y$}

\tlabel(0.25,3.1){\scriptsize $1$}

\tlabel(-0.5,-0.5){\scriptsize $O$}

\tlabel(2,-0.5){\scriptsize $B(1,0)$}

\xmarks{0 step 3 until 3}

\ymarks{0 step 3 until 3}

\polyline{(0,0), (3,5.196)}

\polyline{(3,0), (3,5.196)}

\polyline{(2.75,0), (2.75, 0.25), (3,0.25)}

\polyline{(3,0), (1.5, 2.5981)}

\tlabel(1.75,2.6){\scriptsize $P$}

\tlabel(3.25,5.25){\scriptsize $Q$}

\end{mfpic}

\end{center}

\begin{enumerate}

\item Show that triangle $OPB$ has area $\dfrac{1}{2} \sin(\theta)$.

\item Show that the circular sector $OPB$ with central angle $\theta$ has area $\dfrac{1}{2} \theta$.

\item Show that triangle $OQB$ has area $\dfrac{1}{2} \tan(\theta)$.

\item Comparing areas, show that $\sin(\theta) < \theta < \tan(\theta)$ for $0 < \theta < \dfrac{\pi}{2}.$

\item Use the inequality $\sin(\theta) < \theta$ to show that $\dfrac{\sin(\theta)}{\theta} < 1$ for $0 < \theta < \dfrac{\pi}{2}.$

\item Use the inequality $\theta < \tan(\theta)$ to show that $\cos(\theta) < \dfrac{\sin(\theta)}{\theta}$ for $0 < \theta < \dfrac{\pi}{2}.$ Combine this with the previous part to complete the proof.

\end{enumerate}

\item Show that $\cos(\theta) < \dfrac{\sin(\theta)}{\theta} < 1$ also holds for $-\dfrac{\pi}{2}< \theta < 0$.

\item Explain why the fact that $\tan(\theta) = 3 = \frac{3}{1}$ does not mean $\sin(\theta) = 3$ and $\cos(\theta) = 1$? (See the solution to number \ref{commontanmistake} in Example \ref{circularfunctionsex}.)

\end{enumerate}

\newpage

\subsection{Answers}

\begin{multicols}{3}

\begin{enumerate}

\item $\tan \left( \dfrac{\pi}{4} \right) = 1$ \vphantom{$\dfrac{2\sqrt{3}}{3}$}

\item $\sec \left( \dfrac{\pi}{6} \right) = \dfrac{2\sqrt{3}}{3}$

\item $\csc \left( \dfrac{5\pi}{6} \right) = 2$ \vphantom{$\dfrac{2\sqrt{3}}{3}$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cot \left( \dfrac{4\pi}{3} \right) = \dfrac{\sqrt{3}}{3}$

\item $\tan \left( -\dfrac{11\pi}{6} \right) = \dfrac{\sqrt{3}}{3}$

\item $\sec \left( -\dfrac{3\pi}{2} \right)$ is undefined

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\csc \left( -\dfrac{\pi}{3} \right) = -\dfrac{2\sqrt{3}}{3}$

\item $\cot \left( \dfrac{13\pi}{2} \right) = 0$

\item $\tan \left( 117\pi \right) = 0$ \vphantom{$\dfrac{2\sqrt{3}}{3}$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sec \left( -\dfrac{5\pi}{3} \right) = 2$

\item $\csc \left( 3\pi \right)$ is undefined \vphantom{$\left( -\dfrac{5\pi}{3} \right)$}

\item $\cot \left( -5\pi \right)$ is undefined \vphantom{$\left( -\dfrac{5\pi}{3} \right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\tan \left( \dfrac{31\pi}{2} \right)$ is undefined

\item $\sec \left( \dfrac{\pi}{4} \right) = \sqrt{2}$ \vphantom{$\left( -\dfrac{5\pi}{3} \right)$}

\item $\csc \left( -\dfrac{7\pi}{4} \right) = \sqrt{2}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cot \left( \dfrac{7\pi}{6} \right) = \sqrt{3}$

\item $\tan \left( \dfrac{2\pi}{3} \right) = -\sqrt{3}$

\item $\sec \left( -7\pi \right) = -1$ \vphantom{$\left( -\dfrac{5\pi}{3} \right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\csc \left( \dfrac{\pi}{2} \right) = 1$ \vphantom{$\left( -\dfrac{5\pi}{3} \right)$}

\item $\cot \left( \dfrac{3\pi}{4} \right) = -1$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sin(\theta) = \frac{3}{5}, \cos(\theta) = -\frac{4}{5}, \tan(\theta) = -\frac{3}{4}, \csc(\theta) = \frac{5}{3}, \sec(\theta) = -\frac{5}{4}, \cot(\theta) = -\frac{4}{3}$

\item $\sin(\theta) = -\frac{12}{13}, \cos(\theta) = -\frac{5}{13}, \tan(\theta) = \frac{12}{5}, \csc(\theta) = -\frac{13}{12}, \sec(\theta) = -\frac{13}{5}, \cot(\theta) = \frac{5}{12}$

\item $\sin(\theta) = \frac{24}{25}, \cos(\theta) = \frac{7}{25}, \tan(\theta) = \frac{24}{7}, \csc(\theta) = \frac{25}{24}, \sec(\theta) = \frac{25}{7}, \cot(\theta) = \frac{7}{24}$

\item $\sin(\theta) = \frac{-4\sqrt{3}}{7}, \cos(\theta) = \frac{1}{7}, \tan(\theta) = -4\sqrt{3}, \csc(\theta) = -\frac{7\sqrt{3}}{12}, \sec(\theta) = 7, \cot(\theta) = -\frac{\sqrt{3}}{12}$

\item $\sin(\theta) = -\frac{\sqrt{91}}{10}, \cos(\theta) = -\frac{3}{10}, \tan(\theta) = \frac{\sqrt{91}}{3}, \csc(\theta) = -\frac{10\sqrt{91}}{91}, \sec(\theta) = -\frac{10}{3}, \cot(\theta) = \frac{3\sqrt{91}}{91}$

\item $\sin(\theta) = \frac{\sqrt{530}}{530}, \cos(\theta) = -\frac{23\sqrt{530}}{530}, \tan(\theta) = -\frac{1}{23}, \csc(\theta) = \sqrt{530}, \sec(\theta) = -\frac{\sqrt{530}}{23}, \cot(\theta) = -23$

\item $\sin(\theta) = -\frac{2\sqrt{5}}{5}, \cos(\theta) = \frac{\sqrt{5}}{5}, \tan(\theta) = -2, \csc(\theta) = -\frac{\sqrt{5}}{2}, \sec(\theta) = \sqrt{5}, \cot(\theta) = -\frac{1}{2}$

\item $\sin(\theta) = \frac{\sqrt{15}}{4}, \cos(\theta) = -\frac{1}{4}, \tan(\theta) = -\sqrt{15}, \csc(\theta) = \frac{4\sqrt{15}}{15}, \sec(\theta) = -4, \cot(\theta) = -\frac{\sqrt{15}}{15}$

\item $\sin(\theta) = -\frac{\sqrt{6}}{6}, \cos(\theta) = -\frac{\sqrt{30}}{6}, \tan(\theta) = \frac{\sqrt{5}}{5}, \csc(\theta) = -\sqrt{6}, \sec(\theta) = -\frac{\sqrt{30}}{5}, \cot(\theta) = \sqrt{5}$

\item $\sin(\theta) = \frac{2\sqrt{2}}{3}, \cos(\theta) = \frac{1}{3}, \tan(\theta) = 2\sqrt{2}, \csc(\theta) = \frac{3\sqrt{2}}{4}, \sec(\theta) = 3, \cot(\theta) = \frac{\sqrt{2}}{4}$

\item $\sin(\theta) = \frac{\sqrt{5}}{5}, \cos(\theta) = \frac{2\sqrt{5}}{5}, \tan(\theta) = \frac{1}{2}, \csc(\theta) = \sqrt{5}, \sec(\theta) = \frac{\sqrt{5}}{2}, \cot(\theta) = 2$

\item $\sin(\theta) = \frac{1}{5}, \cos(\theta) = -\frac{2\sqrt{6}}{5}, \tan(\theta) = -\frac{\sqrt{6}}{12}, \csc(\theta) = 5, \sec(\theta) = -\frac{5\sqrt{6}}{12}, \cot(\theta) = -2\sqrt{6}$

\item $\sin(\theta) = -\frac{\sqrt{110}}{11}, \cos(\theta) = -\frac{\sqrt{11}}{11}, \tan(\theta) = \sqrt{10}, \csc(\theta) = -\frac{\sqrt{110}}{10}, \sec(\theta) = -\sqrt{11}, \cot(\theta) = \frac{\sqrt{10}}{10}$

\item $\sin(\theta) = -\frac{\sqrt{95}}{10}, \cos(\theta) = \frac{\sqrt{5}}{10}, \tan(\theta) = -\sqrt{19}, \csc(\theta) = -\frac{2\sqrt{95}}{19}, \sec(\theta) = 2\sqrt{5}, \cot(\theta) = -\frac{\sqrt{19}}{19}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\csc(78.95^{\circ}) \approx 1.019$

\item $\tan(-2.01) \approx 2.129$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cot(392.994) \approx 3.292$

\item $\sec(207^{\circ}) \approx -1.122$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\csc(5.902) \approx -2.688$

\item $\tan(39.672^{\circ}) \approx 0.829$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cot(3^{\circ}) \approx 19.081$

\item $\sec(0.45) \approx 1.111$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\tan(\theta) = \sqrt{3}$ when $\theta = \dfrac{\pi}{3} + \pi k$ for any integer $k$

\item $\sec(\theta) = 2$ when $\theta = \dfrac{\pi}{3} + 2\pi k$ or $\theta = \dfrac{5\pi}{3} + 2\pi k$ for any integer $k$

\item $\csc(\theta) = -1$ when $\theta = \dfrac{3\pi}{2} + 2\pi k$ for any integer $k$.

\item $\cot(\theta) = \dfrac{\sqrt{3}}{3}$ when $\theta = \dfrac{\pi}{3} + \pi k$ for any integer $k$

\item $\tan(\theta) = 0$ when $\theta = \pi k$ for any integer $k$

\item $\sec(\theta) = 1$ when $\theta = 2\pi k$ for any integer $k$

\item $\csc(\theta) = 2$ when $\theta = \dfrac{\pi}{6} + 2\pi k$ or $\theta = \dfrac{5\pi}{6} + 2\pi k$ for any integer $k$.

\item $\cot(\theta) = 0$ when $\theta = \dfrac{\pi}{2} + \pi k$ for any integer $k$

\item $\tan(\theta) = -1$ when $\theta = \dfrac{3\pi}{4} + \pi k$ for any integer $k$

\item $\sec(\theta) = 0$ never happens

\item $\csc(\theta) = -\dfrac{1}{2}$ never happens

\item $\sec(\theta) = -1$ when $\theta = \pi + 2\pi k = (2k+1)\pi$ for any integer $k$

\item $\tan(\theta) = -\sqrt{3}$ when $\theta = \dfrac{2\pi}{3} + \pi k$ for any integer $k$

\item $\csc(\theta) = -2$ when $\theta = \dfrac{7\pi}{6} + 2\pi k$ or $\theta = \dfrac{11\pi}{6} + 2\pi k$ for any integer $k$

\item $\cot(\theta) = -1$ when $\theta = \dfrac{3\pi}{4} + \pi k$ for any integer $k$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cot(t) = 1$ when $t = \dfrac{\pi}{4} + \pi k$ for any integer $k$

\item $\tan(t) = \dfrac{\sqrt{3}}{3}$ when $t = \dfrac{\pi}{6} + \pi k$ for any integer $k$

\item $\sec(t) = -\dfrac{2\sqrt{3}}{3}$ when $t = \dfrac{5\pi}{6} + 2\pi k$ or $t = \dfrac{7\pi}{6} + 2\pi k$ for any integer $k$

\item $\csc(t) = 0$ never happens

\item $\cot(t) = -\sqrt{3}$ when $t = \dfrac{5\pi}{6} + \pi k$ for any integer $k$

\item $\tan(t) = -\dfrac{\sqrt{3}}{3}$ when $t = \dfrac{5\pi}{6} + \pi k$ for any integer $k$

\item $\sec(t) = \dfrac{2\sqrt{3}}{3}$ when $t = \dfrac{\pi}{6} + 2\pi k$ or $t = \dfrac{11\pi}{6} + 2\pi k$ for any integer $k$

\item $\csc(t) = \dfrac{2\sqrt{3}}{3}$ when $t = \dfrac{\pi}{3} + 2\pi k$ or $t = \dfrac{2\pi}{3} + 2\pi k$ for any integer $k$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\theta = 30^{\circ}$, $a = 3\sqrt{3}$, $c = \sqrt{108} = 6\sqrt{3}$

\item $\alpha = 56^{\circ}$, $b = 12 \tan(34^{\circ}) = 8.094$, $c = 12\sec(34^{\circ}) = \dfrac{12}{\cos(34^{\circ})} \approx 14.475$

\item $\theta = 43^{\circ}$, $a = 6\cot(47^{\circ}) = \dfrac{6}{\tan(47^{\circ})} \approx 5.595$, $c = 6\csc(47^{\circ}) = \dfrac{6}{\sin(47^{\circ})} \approx 8.204$

\item $\beta = 40^{\circ}$, $b = 2.5 \tan(50^{\circ}) \approx 2.979$, $c = 2.5\sec(50^{\circ}) = \dfrac{2.5}{\cos(50^{\circ})} \approx 3.889$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item The side adjacent to $\theta$ has length $4\sqrt{3} \approx 6.928$

\item The side opposite $\theta$ has length $10 \sin(15^{\circ}) \approx 2.588$

\item The side opposite $\theta$ is $2\tan(87^{\circ}) \approx 38.162$

\item The hypoteneuse has length $14 \csc(38.2^{\circ}) = \dfrac{14}{\sin(38.2^{\circ})} \approx 22.639$

\item The side adjacent to $\theta$ has length $3.98 \cos(2.05^{\circ}) \approx 3.977$

\item The side opposite $\theta$ has length $31\tan(42^{\circ}) \approx 27.912$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item The tree is about 47 feet tall.

\item The lights are about 75 feet apart.

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \begin{enumerate}

\addtocounter{enumii}{1}

\item The fire is about 4581 feet from the base of the tower.

\item The Sasquatch ran $200\cot(6^{\circ}) - 200\cot(6.5^{\circ}) \approx 147$ feet in those 10 seconds. This translates to $\approx 10$ miles per hour. At the scene of the second sighting, the Sasquatch was $\approx 1755$ feet from the tower, which means, if it keeps up this pace, it will reach the tower in about $2$ minutes.

\end{enumerate}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item The tree is about 41 feet tall.

\item The boat has traveled about 244 feet.

\item The tower is about 682 feet tall. The guy wire hits the ground about 731 feet away from the base of the tower.

\end{enumerate}

\closegraphsfile

10.4: Trigonometric Identities

\subsection{Exercises}

In Exercises \ref{evenoddfirst} - \ref{evenoddlast}, use the Even / Odd Identities to verify the identity. Assume all quantities are defined.

\begin{multicols}{2}

\begin{enumerate}

\item $\sin(3\pi - 2\theta) = -\sin(2\theta - 3\pi)$ \vphantom{$\left( -\dfrac{\pi}{4} \right)$} \label{evenoddfirst}

\item $\cos \left( -\dfrac{\pi}{4} - 5t \right) = \cos \left( 5t + \dfrac{\pi}{4} \right)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\tan(-t^{2} + 1) = -\tan(t^{2} - 1)$

\item $\csc(-\theta - 5) = -\csc(\theta + 5)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sec(-6t) = \sec(6t)$

\item $\cot(9 - 7\theta) = -\cot(7\theta - 9)$ \label{evenoddlast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

In Exercises \ref{sumdifffirst} - \ref{sumdifflast}, use the Sum and Difference Identities to find the exact value. You may have need of the Quotient, Reciprocal or Even / Odd Identities as well.

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \label{cos75} $\cos(75^{\circ})$ \label{sumdifffirst}

\item $\sec(165^{\circ})$

\item \label{sin105} $\sin(105^{\circ})$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\csc(195^{\circ})$

\item $\cot(255^{\circ})$

\item $\tan(375^{\circ})$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos\left(\dfrac{13\pi}{12}\right)$

\item $\sin\left(\dfrac{11\pi}{12}\right)$

\item $\tan\left(\dfrac{13\pi}{12}\right)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \label{cos7pi12} $\cos \left( \dfrac{7\pi}{12} \right)$

\item $\tan \left( \dfrac{17\pi}{12} \right)$

\item \label{sinpi12} $\sin \left( \dfrac{\pi}{12} \right)$ \vphantom{$\left(\dfrac{13\pi}{12}\right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cot \left( \dfrac{11\pi}{12} \right)$

\item $\csc \left( \dfrac{5\pi}{12} \right)$

\item $\sec \left( -\dfrac{\pi}{12} \right)$ \vphantom{$\left(\dfrac{13\pi}{12}\right)$} \label{sumdifflast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item If $\alpha$ is a Quadrant IV angle with $\cos(\alpha) = \dfrac{\sqrt{5}}{5}$, and $\sin(\beta) = \dfrac{\sqrt{10}}{10}$, where $\dfrac{\pi}{2} < \beta < \pi$, find

\begin{multicols}{3}

\begin{enumerate}

\item $\cos(\alpha + \beta)$

\item $\sin(\alpha + \beta)$

\item $\tan(\alpha + \beta)$

\setcounter{HWindent}{\value{enumii}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumii}{\value{HWindent}}

\item $\cos(\alpha - \beta)$

\item $\sin(\alpha - \beta)$

\item $\tan(\alpha - \beta)$

\end{enumerate}

\end{multicols}

\item If $\csc(\alpha) = 3$, where $0 < \alpha < \dfrac{\pi}{2}$, and $\beta$ is a Quadrant II angle with $\tan(\beta) = -7$, find

\begin{multicols}{3}

\begin{enumerate}

\item $\cos(\alpha + \beta)$

\item $\sin(\alpha + \beta)$

\item $\tan(\alpha + \beta)$

\setcounter{HWindent}{\value{enumii}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumii}{\value{HWindent}}

\item $\cos(\alpha - \beta)$

\item $\sin(\alpha - \beta)$

\item $\tan(\alpha - \beta)$

\end{enumerate}

\end{multicols}

\item If $\sin(\alpha) = \dfrac{3}{5}$, where $0 < \alpha < \dfrac{\pi}{2}$, and $\cos(\beta) = \dfrac{12}{13}$ where $\dfrac{3\pi}{2} < \beta < 2\pi$, find

\begin{multicols}{3}

\begin{enumerate}

\item $\sin(\alpha + \beta)$

\item $\cos(\alpha - \beta)$

\item $\tan(\alpha - \beta)$

\end{enumerate}

\end{multicols}

\pagebreak

\item If $\sec(\alpha) = -\dfrac{5}{3}$, where $\dfrac{\pi}{2} < \alpha < \pi$, and $\tan(\beta) = \dfrac{24}{7}$, where $\pi < \beta < \dfrac{3\pi}{2}$, find

\begin{multicols}{3}

\begin{enumerate}

\item $\csc(\alpha - \beta)$

\item $\sec(\alpha + \beta)$

\item $\cot(\alpha + \beta)$

\end{enumerate}

\end{multicols}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

In Exercises \ref{identfirstident} - \ref{identlastident}, verify the identity.

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos(\theta - \pi) = -\cos(\theta)$ \label{identfirstident}

\item $\sin(\pi - \theta) = \sin(\theta)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\tan\left(\theta + \dfrac{\pi}{2} \right) = -\cot(\theta)$

\item $\sin(\alpha + \beta) + \sin(\alpha - \beta) = 2\sin(\alpha)\cos(\beta)$ \vphantom{$\left( \dfrac{\pi}{2} \right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sin(\alpha + \beta) - \sin(\alpha - \beta) = 2\cos(\alpha) \sin(\beta)$

\item $\cos(\alpha + \beta) + \cos(\alpha - \beta) = 2\cos(\alpha) \cos(\beta)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos(\alpha + \beta) - \cos(\alpha - \beta) = -2\sin(\alpha) \sin(\beta)$ \vphantom{$\dfrac{\sin(\alpha+\beta)}{\sin(\alpha-\beta)}$}

\item $\dfrac{\sin(\alpha+\beta)}{\sin(\alpha-\beta)} = \dfrac{1+\cot(\alpha) \tan(\beta)}{1 - \cot(\alpha) \tan(\beta)}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\dfrac{\cos(\alpha + \beta)}{\cos(\alpha - \beta)} = \dfrac{1 - \tan(\alpha)\tan(\beta)}{1 + \tan(\alpha)\tan(\beta)}$

\item $\dfrac{\tan(\alpha + \beta)}{\tan(\alpha - \beta)} = \dfrac{\sin(\alpha)\cos(\alpha) + \sin(\beta)\cos(\beta)}{\sin(\alpha)\cos(\alpha) - \sin(\beta)\cos(\beta)}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\dfrac{\sin(t + h) - \sin(t)}{h} = \cos(t) \left(\dfrac{\sin(h)}{h} \right) + \sin(t) \left( \dfrac{\cos(h) - 1}{h} \right)$

\item $\dfrac{\cos(t + h) - \cos(t)}{h} = \cos(t) \left( \dfrac{\cos(h) - 1}{h} \right) - \sin(t) \left(\dfrac{\sin(h)}{h} \right)$

\item $\dfrac{\tan(t + h) - \tan(t)}{h} = \left( \dfrac{\tan(h)}{h} \right) \left(\dfrac{\sec^{2}(t)}{1 - \tan(t)\tan(h)} \right)$ \label{identlastident}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

In Exercises \ref{idenhalfanglefirst} - \ref{idenhalfanglelast}, use the Half Angle Formulas to find the exact value. You may have need of the Quotient, Reciprocal or Even / Odd Identities as well.

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos(75^{\circ})$ (compare with Exercise \ref{cos75}) \label{idenhalfanglefirst}

\item $\sin(105^{\circ})$ (compare with Exercise \ref{sin105})

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos(67.5^{\circ})$

\item $\sin(157.5^{\circ})$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\tan(112.5^{\circ})$ \vphantom{$\left( \dfrac{7\pi}{12} \right)$}

\item $\cos\left( \dfrac{7\pi}{12} \right)$ (compare with Exercise \ref{cos7pi12})

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sin\left( \dfrac{\pi}{12} \right)$ (compare with Exercise \ref{sinpi12})

\item $\cos \left( \dfrac{\pi}{8} \right)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sin \left( \dfrac{5\pi}{8} \right)$

\item $\tan \left( \dfrac{7\pi}{8} \right)$ \label{idenhalfanglelast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\pagebreak

In Exercises \ref{doublehalffirst} - \ref{doublehalflast}, use the given information about $\theta$ to find the exact values of

\begin{multicols}{3}

\begin{itemize}

\item $\sin(2\theta)$

\item $\sin\left(\dfrac{\theta}{2}\right)$

\item $\cos(2\theta)$

\item $\cos\left(\dfrac{\theta}{2}\right)$

\item $\tan(2\theta)$

\item $\tan\left(\dfrac{\theta}{2}\right)$

\end{itemize}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sin(\theta) = -\dfrac{7}{25}$ where $\dfrac{3\pi}{2} < \theta < 2\pi$ \label{doublehalffirst}

\item $\cos(\theta) = \dfrac{28}{53}$ where $0 < \theta < \dfrac{\pi}{2}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\tan(\theta) = \dfrac{12}{5}$ where $\pi < \theta < \dfrac{3\pi}{2}$

\item $\csc(\theta) = 4$ where $\dfrac{\pi}{2} < \theta < \pi$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos(\theta) = \dfrac{3}{5}$ where $0 < \theta < \dfrac{\pi}{2}$

\item $\sin(\theta) = -\dfrac{4}{5}$ where $\pi < \theta < \dfrac{3\pi}{2}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos(\theta) = \dfrac{12}{13}$ where $\dfrac{3\pi}{2} < \theta < 2\pi$

\item $\sin(\theta) = \dfrac{5}{13}$ where $\dfrac{\pi}{2} < \theta < \pi$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sec(\theta) = \sqrt{5}$ where $\dfrac{3\pi}{2} < \theta < 2\pi$

\item $\tan(\theta) = -2$ where $\dfrac{\pi}{2} < \theta < \pi$ \label{doublehalflast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

In Exercises \ref{moreidentfirst} - \ref{moreidentlast}, verify the identity. Assume all quantities are defined.

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $(\cos(\theta) + \sin(\theta))^2 = 1 + \sin(2\theta)$ \label{moreidentfirst}

\item $(\cos(\theta) - \sin(\theta))^2 = 1 - \sin(2\theta)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\tan(2\theta) = \dfrac{1}{1-\tan(\theta)} - \dfrac{1}{1+\tan(\theta)}$

\item $\csc(2\theta) = \dfrac{\cot(\theta) + \tan(\theta)}{2}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $8 \sin^{4}(\theta) = \cos(4\theta) - 4\cos(2\theta)+3$

\item $8 \cos^{4}(\theta) = \cos(4\theta) + 4\cos(2\theta)+3$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \label{sine3theta} $\sin(3\theta) = 3\sin(\theta) - 4\sin^{3}(\theta)$

\item $\sin(4\theta) = 4\sin(\theta)\cos^{3}(\theta) - 4\sin^{3}(\theta)\cos(\theta)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $32\sin^{2}(\theta) \cos^{4}(\theta) = 2 + \cos(2\theta) - 2\cos(4\theta) - \cos(6\theta)$

\item $32\sin^{4}(\theta) \cos^{2}(\theta) = 2 - \cos(2\theta) - 2\cos(4\theta) + \cos(6\theta)$

\item \label{cosine4theta} $\cos(4\theta) = 8\cos^{4}(\theta) - 8\cos^{2}(\theta) + 1$

\item $\cos(8\theta) = 128\cos^{8}(\theta)-256\cos^{6}(\theta)+160\cos^{4}(\theta)-32\cos^{2}(\theta)+1$ (HINT: Use the result to \ref{cosine4theta}.)

\item $\sec(2\theta) = \dfrac{\cos(\theta)}{\cos(\theta) + \sin(\theta)} + \dfrac{\sin(\theta)}{\cos(\theta)-\sin(\theta)}$

\item $\dfrac{1}{\cos(\theta) - \sin(\theta)} + \dfrac{1}{\cos(\theta) + \sin(\theta)} = \dfrac{2\cos(\theta)}{\cos(2\theta)}$

\item $\dfrac{1}{\cos(\theta) - \sin(\theta)} - \dfrac{1}{\cos(\theta) + \sin(\theta)} = \dfrac{2\sin(\theta)}{\cos(2\theta)}$ \label{moreidentlast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\pagebreak

In Exercises \ref{prodsumfirst} - \ref{prodsumlast}, write the given product as a sum. You may need to use an Even/Odd Identity.

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos(3\theta)\cos(5\theta)$ \label{prodsumfirst}

\item $\sin(2\theta)\sin(7\theta)$

\item $\sin(9\theta)\cos(\theta)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos(2\theta) \cos(6\theta)$

\item $\sin(3\theta) \sin(2\theta)$

\item $\cos(\theta) \sin(3\theta)$ \label{prodsumlast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

In Exercises \ref{sumprodfirst} - \ref{sumprodlast}, write the given sum as a product. You may need to use an Even/Odd or Cofunction Identity.

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos(3\theta) + \cos(5\theta)$ \label{sumprodfirst}

\item $\sin(2\theta) - \sin(7\theta)$

\item $\cos(5\theta) - \cos(6\theta)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sin(9\theta) - \sin(-\theta)$

\item $\sin(\theta) + \cos(\theta)$

\item $\cos(\theta) - \sin(\theta)$ \label{sumprodlast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \label{preludetoarctrigsine} Suppose $\theta$ is a Quadrant I angle with $\sin(\theta) = x$. Verify the following formulas

\begin{multicols}{3}

\begin{enumerate}

\item $\cos(\theta) = \sqrt{1-x^2}$

\item $\sin(2\theta) = 2x\sqrt{1-x^2}$

\item $\cos(2\theta) = 1 - 2x^2$

\end{enumerate}

\end{multicols}

\item Discuss with your classmates how each of the formulas, if any, in Exercise \ref{preludetoarctrigsine} change if we change assume $\theta$ is a Quadrant II, III, or IV angle.

\item \label{preludetoarctrigtan} Suppose $\theta$ is a Quadrant I angle with $\tan(\theta) = x$. Verify the following formulas

\begin{multicols}{2}

\begin{enumerate}

\item $\cos(\theta) = \dfrac{1}{\sqrt{x^2+1}}$

\item $\sin(\theta) = \dfrac{x}{\sqrt{x^2+1}}$

\setcounter{HWindent}{\value{enumii}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumii}{\value{HWindent}}

\item $\sin(2\theta) = \dfrac{2x}{x^2+1}$

\item $\cos(2\theta) = \dfrac{1-x^2}{x^2+1}$

\end{enumerate}

\end{multicols}

\item Discuss with your classmates how each of the formulas, if any, in Exercise \ref{preludetoarctrigtan} change if we change assume $\theta$ is a Quadrant II, III, or IV angle.

\item If $\sin(\theta) = \dfrac{x}{2}$ for $-\dfrac{\pi}{2} < \theta < \dfrac{\pi}{2}$, find an expression for $\cos(2\theta)$ in terms of $x$.

\item If $\tan(\theta) = \dfrac{x}{7}$ for $-\dfrac{\pi}{2} < \theta < \dfrac{\pi}{2}$, find an expression for $\sin(2\theta)$ in terms of $x$.

\item If $\sec(\theta) = \dfrac{x}{4}$ for $0 < \theta < \dfrac{\pi}{2}$, find an expression for $\ln|\sec(\theta) + \tan(\theta)|$ in terms of $x$.

\item Show that $\cos^{2}(\theta) - \sin^{2}(\theta) = 2\cos^{2}(\theta) - 1 = 1 - 2\sin^{2}(\theta)$ for all $\theta$.

\item Let $\theta$ be a Quadrant III angle with $\cos(\theta) = -\dfrac{1}{5}$. Show that this is not enough information to determine the sign of $\sin\left(\dfrac{\theta}{2}\right)$ by first assuming $3\pi < \theta < \dfrac{7\pi}{2}$ and then assuming $\pi < \theta < \dfrac{3\pi}{2}$ and computing $\sin\left(\dfrac{\theta}{2}\right)$ in both cases.

\item Without using your calculator, show that $\dfrac{\sqrt{2 + \sqrt{3}}}{2} = \dfrac{\sqrt{6} + \sqrt{2}}{4}$

\item In part \ref{cosinepolynomial} of Example \ref{doubleangleex}, we wrote $\cos(3\theta)$ as a polynomial in terms of $\cos(\theta)$. In Exercise \ref{cosine4theta}, we had you verify an identity which expresses $\cos(4\theta)$ as a polynomial in terms of $\cos(\theta)$. Can you find a polynomial in terms of $\cos(\theta)$ for $\cos(5\theta)$? $\cos(6\theta)$? Can you find a pattern so that $\cos(n\theta)$ could be written as a polynomial in cosine for any natural number $n$?

\item In Exercise \ref{sine3theta}, we has you verify an identity which expresses $\sin(3\theta)$ as a polynomial in terms of $\sin(\theta)$. Can you do the same for $\sin(5\theta)$? What about for $\sin(4\theta)$? If not, what goes wrong?

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item Verify the Even / Odd Identities for tangent, secant, cosecant and cotangent.

\item Verify the Cofunction Identities for tangent, secant, cosecant and cotangent.

\item Verify the Difference Identities for sine and tangent.

\item Verify the Product to Sum Identities.

\item Verify the Sum to Product Identities.

\end{enumerate}

\newpage

\subsection{Answers}

\begin{multicols}{2}

\begin{enumerate}

\addtocounter{enumi}{6}

\item $\cos(75^{\circ}) = \dfrac{\sqrt{6} - \sqrt{2}}{4} $

\item $\sec(165^{\circ}) = -\dfrac{4}{\sqrt{2}+\sqrt{6}} = \sqrt{2} - \sqrt{6}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sin(105^{\circ}) = \dfrac{\sqrt{6}+\sqrt{2}}{4}$

\item $\csc(195^{\circ}) = \dfrac{4}{\sqrt{2}-\sqrt{6}} = -(\sqrt{2}+\sqrt{6})$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cot(255^{\circ}) = \dfrac{\sqrt{3}-1}{\sqrt{3}+1} = 2-\sqrt{3}$

\item $\tan(375^{\circ}) = \dfrac{3-\sqrt{3}}{3+\sqrt{3}} = 2-\sqrt{3}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos\left(\dfrac{13\pi}{12}\right) = -\dfrac{\sqrt{6}+\sqrt{2}}{4}$

\item $\sin\left(\dfrac{11\pi}{12}\right) = \dfrac{\sqrt{6} - \sqrt{2}}{4}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\tan\left(\dfrac{13\pi}{12}\right) = \dfrac{3-\sqrt{3}}{3+\sqrt{3}} = 2-\sqrt{3}$

\item $\cos \left( \dfrac{7\pi}{12} \right) = \dfrac{\sqrt{2} - \sqrt{6}}{4}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\tan \left( \dfrac{17\pi}{12} \right) = 2 + \sqrt{3}$

\item $\sin \left( \dfrac{\pi}{12} \right) = \dfrac{\sqrt{6} - \sqrt{2}}{4}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cot \left( \dfrac{11\pi}{12} \right) = -(2 + \sqrt{3})$

\item $\csc \left( \dfrac{5\pi}{12} \right) = \sqrt{6} - \sqrt{2}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sec \left( -\dfrac{\pi}{12} \right) = \sqrt{6} - \sqrt{2}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \begin{multicols}{2}

\begin{enumerate}

\item $\cos(\alpha + \beta) = -\dfrac{\sqrt{2}}{10}$

\item $\sin(\alpha + \beta) = \dfrac{7\sqrt{2}}{10}$

\setcounter{HWindent}{\value{enumii}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumii}{\value{HWindent}}

\item $\tan(\alpha + \beta) = -7$ \vphantom{$\dfrac{\sqrt{2}}{2}$}

\item $\cos(\alpha - \beta)= -\dfrac{\sqrt{2}}{2}$

\setcounter{HWindent}{\value{enumii}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumii}{\value{HWindent}}

\item $\sin(\alpha - \beta) = \dfrac{\sqrt{2}}{2}$

\item $\tan(\alpha - \beta) = -1$ \vphantom{$\dfrac{\sqrt{2}}{2}$}

\end{enumerate}

\end{multicols}

\item \begin{multicols}{2}

\begin{enumerate}

\item $\cos(\alpha + \beta) = - \dfrac{4+7\sqrt{2}}{30}$

\item $\sin(\alpha + \beta) = \dfrac{28-\sqrt{2}}{30}$

\setcounter{HWindent}{\value{enumii}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumii}{\value{HWindent}}

\item $\tan(\alpha + \beta) = \dfrac{-28+\sqrt{2}}{4+7\sqrt{2}} = \dfrac{63-100\sqrt{2}}{41}$

\item $\cos(\alpha - \beta) = \dfrac{-4+7\sqrt{2}}{30}$

\setcounter{HWindent}{\value{enumii}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumii}{\value{HWindent}}

\item $\sin(\alpha - \beta) = - \dfrac{28+\sqrt{2}}{30}$

\item $\tan(\alpha - \beta)= \dfrac{28+\sqrt{2}}{4-7\sqrt{2}} = -\dfrac{63+100\sqrt{2}}{41}$

\end{enumerate}

\end{multicols}

\item \begin{multicols}{3}

\begin{enumerate}

\item $\sin(\alpha + \beta) = \dfrac{16}{65}$

\item $\cos(\alpha - \beta) = \dfrac{33}{65}$

\item $\tan(\alpha - \beta) = \dfrac{56}{33}$

\end{enumerate}

\end{multicols}

\pagebreak

\item \begin{multicols}{3}

\begin{enumerate}

\item $\csc(\alpha - \beta) = -\dfrac{5}{4}$

\item $\sec(\alpha + \beta) = \dfrac{125}{117}$

\item $\cot(\alpha + \beta) = \dfrac{117}{44}$

\end{enumerate}

\end{multicols}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\addtocounter{enumi}{13}

\item $\cos(75^{\circ}) = \dfrac{\sqrt{2-\sqrt{3}}}{2}$

\item $\sin(105^{\circ}) = \dfrac{\sqrt{2+\sqrt{3}}}{2}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos(67.5^{\circ}) = \dfrac{\sqrt{2-\sqrt{2}}}{2}$

\item $\sin(157.5^{\circ}) = \dfrac{\sqrt{2-\sqrt{2}}}{2}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\tan(112.5^{\circ}) = - \sqrt{\dfrac{2+\sqrt{2}}{2-\sqrt{2}}} = -1 - \sqrt{2}$

\item $\cos\left( \dfrac{7\pi}{12} \right) = -\dfrac{\sqrt{2-\sqrt{3}}}{2}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sin\left( \dfrac{\pi}{12} \right) = \dfrac{\sqrt{2-\sqrt{3}}}{2}$

\item $\cos \left( \dfrac{\pi}{8} \right) = \dfrac{\sqrt{2 + \sqrt{2}}}{2}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sin \left( \dfrac{5\pi}{8} \right) = \dfrac{\sqrt{2 + \sqrt{2}}}{2}$

\item $\tan \left( \dfrac{7\pi}{8} \right) = -\sqrt{ \dfrac{2 - \sqrt{2}}{2 + \sqrt{2}} } =1-\sqrt{2}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \begin{multicols}{3}

\begin{itemize}

\item $\sin(2\theta) = -\dfrac{336}{625}$

\item $\sin\left(\frac{\theta}{2}\right) = \dfrac{\sqrt{2}}{10}$

\item $\cos(2\theta) = \dfrac{527}{625}$

\item $\cos\left(\frac{\theta}{2}\right) = -\dfrac{7\sqrt{2}}{10}$

\item $\tan(2\theta) = -\dfrac{336}{527}$

\item $\tan\left(\frac{\theta}{2}\right) = -\dfrac{1}{7}$

\end{itemize}

\end{multicols}

\item \begin{multicols}{3}

\begin{itemize}

\item $\sin(2\theta) = \dfrac{2520}{2809}$

\item $\sin\left(\frac{\theta}{2}\right) = \dfrac{5\sqrt{106}}{106}$

\item $\cos(2\theta) = -\dfrac{1241}{2809}$

\item $\cos\left(\frac{\theta}{2}\right) = \dfrac{9\sqrt{106}}{106}$

\item $\tan(2\theta) = -\dfrac{2520}{1241}$

\item $\tan\left(\frac{\theta}{2}\right) = \dfrac{5}{9}$

\end{itemize}

\end{multicols}

\item \begin{multicols}{3}

\begin{itemize}

\item $\sin(2\theta) = \dfrac{120}{169}$

\item $\sin\left(\frac{\theta}{2}\right) = \dfrac{3\sqrt{13}}{13}$

\item $\cos(2\theta) = -\dfrac{119}{169}$

\item $\cos\left(\frac{\theta}{2}\right) = -\dfrac{2\sqrt{13}}{13}$

\item $\tan(2\theta) = -\dfrac{120}{119}$

\item $\tan\left(\frac{\theta}{2}\right) = -\dfrac{3}{2}$

\end{itemize}

\end{multicols}

\item \begin{multicols}{3}

\begin{itemize}

\item $\sin(2\theta) = -\dfrac{\sqrt{15}}{8}$

\item $\sin\left(\frac{\theta}{2}\right) =\dfrac{\sqrt{8+2\sqrt{15}}}{4} \\ \phantom{\tan\left(\frac{\theta}{2}\right) = 4+\sqrt{15}}$

\item $\cos(2\theta) = \dfrac{7}{8}$

\item $\cos\left(\frac{\theta}{2}\right) = \dfrac{\sqrt{8-2\sqrt{15}}}{4} \\ \phantom{\tan\left(\frac{\theta}{2}\right) = 4+\sqrt{15}} $

\item $\tan(2\theta) = -\dfrac{\sqrt{15}}{7}$

\item $\tan\left(\frac{\theta}{2}\right) = \sqrt{\dfrac{8+2\sqrt{15}}{8-2\sqrt{15}}} \\ \tan\left(\frac{\theta}{2}\right) = 4+\sqrt{15}$

\end{itemize}

\end{multicols}

\item \begin{multicols}{3}

\begin{itemize}

\item $\sin(2\theta) = \dfrac{24}{25}$

\item $\sin\left(\frac{\theta}{2}\right) = \dfrac{\sqrt{5}}{5}$

\item $\cos(2\theta) = -\dfrac{7}{25}$

\item $\cos\left(\frac{\theta}{2}\right) = \dfrac{2\sqrt{5}}{5}$

\item $\tan(2\theta)=-\dfrac{24}{7} $

\item $\tan\left(\frac{\theta}{2}\right) = \dfrac{1}{2}$

\end{itemize}

\end{multicols}

\pagebreak

\item \begin{multicols}{3}

\begin{itemize}

\item $\sin(2\theta) = \dfrac{24}{25}$

\item $\sin\left(\frac{\theta}{2}\right) = \dfrac{2\sqrt{5}}{5}$

\item $\cos(2\theta) = -\dfrac{7}{25}$

\item $\cos\left(\frac{\theta}{2}\right) = -\dfrac{\sqrt{5}}{5}$

\item $\tan(2\theta)=-\dfrac{24}{7} $

\item $\tan\left(\frac{\theta}{2}\right) = -2$

\end{itemize}

\end{multicols}

\item \begin{multicols}{3}

\begin{itemize}

\item $\sin(2\theta) = -\dfrac{120}{169}$

\item $\sin\left(\frac{\theta}{2}\right) = \dfrac{\sqrt{26}}{26}$

\item $\cos(2\theta) = \dfrac{119}{169}$

\item $\cos\left(\frac{\theta}{2}\right) = -\dfrac{5\sqrt{26}}{26}$

\item $\tan(2\theta)=-\dfrac{120}{119}$

\item $\tan\left(\frac{\theta}{2}\right) = -\dfrac{1}{5}$

\end{itemize}

\end{multicols}

\item \begin{multicols}{3}

\begin{itemize}

\item $\sin(2\theta) = -\dfrac{120}{169}$

\item $\sin\left(\frac{\theta}{2}\right) = \dfrac{5\sqrt{26}}{26}$

\item $\cos(2\theta) = \dfrac{119}{169}$

\item $\cos\left(\frac{\theta}{2}\right) = \dfrac{\sqrt{26}}{26}$

\item $\tan(2\theta)=-\dfrac{120}{119}$

\item $\tan\left(\frac{\theta}{2}\right) = 5$

\end{itemize}

\end{multicols}

\item \begin{multicols}{3}

\begin{itemize}

\item $\sin(2\theta) = -\dfrac{4}{5}$

\item $\sin\left(\frac{\theta}{2}\right) = \dfrac{\sqrt{50-10\sqrt{5}}}{10} \\ \phantom{\tan\left(\frac{\theta}{2}\right) =\dfrac{5-5\sqrt{5}}{10}}$

\item $\cos(2\theta) = -\dfrac{3}{5}$

\item $\cos\left(\frac{\theta}{2}\right)= -\dfrac{\sqrt{50+10\sqrt{5}}}{10} \\ \phantom{\tan\left(\frac{\theta}{2}\right) =\dfrac{5-5\sqrt{5}}{10}}$

\item $\tan(2\theta)=\dfrac{4}{3}$

\item $\tan\left(\frac{\theta}{2}\right) = -\sqrt{\dfrac{5-\sqrt{5}}{5+\sqrt{5}}} \\ \tan\left(\frac{\theta}{2}\right) =\dfrac{5-5\sqrt{5}}{10}$

\end{itemize}

\end{multicols}

\item \begin{multicols}{3}

\begin{itemize}

\item $\sin(2\theta) = -\dfrac{4}{5}$

\item $\sin\left(\frac{\theta}{2}\right) = \dfrac{\sqrt{50+10\sqrt{5}}}{10} \\ \phantom{\tan\left(\frac{\theta}{2}\right) =\dfrac{5-5\sqrt{5}}{10}}$

\item $\cos(2\theta) = -\dfrac{3}{5}$

\item $\cos\left(\frac{\theta}{2}\right)= \dfrac{\sqrt{50-10\sqrt{5}}}{10} \\ \phantom{\tan\left(\frac{\theta}{2}\right) =\dfrac{5-5\sqrt{5}}{10}}$

\item $\tan(2\theta)=\dfrac{4}{3}$

\item $\tan\left(\frac{\theta}{2}\right) = \sqrt{\dfrac{5+\sqrt{5}}{5-\sqrt{5}}} \\ \tan\left(\frac{\theta}{2}\right) =\dfrac{5+5\sqrt{5}}{10}$

\end{itemize}

\end{multicols}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\addtocounter{enumi}{15}

\item $\dfrac{\cos(2\theta) + \cos(8\theta)}{2}$

\item $\dfrac{\cos(5\theta) - \cos(9\theta)}{2}$

\item $\dfrac{\sin(8\theta) + \sin(10\theta)}{2}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\dfrac{\cos(4\theta) + \cos(8\theta)}{2}$

\item $\dfrac{\cos(\theta) - \cos(5\theta)}{2}$

\item $\dfrac{\sin(2\theta) + \sin(4\theta)}{2}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $2\cos(4\theta)\cos(\theta)$

\item $-2\cos \left( \dfrac{9}{2}\theta \right) \sin \left( \dfrac{5}{2}\theta \right)$

\item $2\sin \left( \dfrac{11}{2}\theta \right) \sin \left( \dfrac{1}{2}\theta \right)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $2\cos(4\theta)\sin(5\theta)$

\item $\sqrt{2}\cos \left(\theta - \dfrac{\pi}{4} \right)$

\item $-\sqrt{2}\sin \left(\theta - \dfrac{\pi}{4} \right)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\addtocounter{enumi}{4}

\item $1 - \dfrac{x^{2}}{2}$

\item $\dfrac{14x}{x^{2} + 49}$

\item $\ln |x + \sqrt{x^{2} + 16}| - \ln(4)$ \vphantom{$\dfrac{14x}{x^{2} + 49}$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\closegraphsfile

10.5: Graphs of the Trigonometric Functions

\subsection{Exercises}

In Exercises \ref{sinecosinegraphfirst} - \ref{sinecosinegraphlast}, graph one cycle of the given function. State the period, amplitude, phase shift and vertical shift of the function.

\begin{multicols}{3}

\begin{enumerate}

\item $y = 3\sin(x)$ \label{sinecosinegraphfirst}

\item $y = \sin(3x)$

\item $y = -2\cos(x)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $y = \cos \left( x - \dfrac{\pi}{2} \right)$

\item $y = -\sin \left( x + \dfrac{\pi}{3} \right)$

\item $y = \sin(2x - \pi)$ \vphantom{$\left( \dfrac{\pi}{2} \right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $y = -\dfrac{1}{3}\cos \left( \dfrac{1}{2}x + \dfrac{\pi}{3} \right)$

\item $y = \cos (3x - 2\pi) + 4$ \vphantom{$\left( \dfrac{1\pi}{2} \right)$}

\item $y = \sin \left( -x - \dfrac{\pi}{4} \right) - 2$ \vphantom{$\left( \dfrac{1\pi}{2} \right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $y = \dfrac{2}{3} \cos \left( \dfrac{\pi}{2} - 4x \right) + 1$

\item $y = -\dfrac{3}{2} \cos \left( 2x + \dfrac{\pi}{3} \right) - \dfrac{1}{2}$

\item $y = 4\sin (-2\pi x + \pi)$ \vphantom{$\left( \dfrac{1\pi}{2} \right)$} \label{sinecosinegraphlast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

In Exercises \ref{othergraphsfirst} - \ref{othergraphslast}, graph one cycle of the given function. State the period of the function.

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $y = \tan \left(x - \dfrac{\pi}{3} \right)$ \vphantom{$\left( \dfrac{1\pi}{2} \right)$} \label{othergraphsfirst}

\item $y = 2\tan \left( \dfrac{1}{4}x \right) - 3$

\item $y = \dfrac{1}{3}\tan(-2x - \pi) + 1$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $y = \sec \left( x - \dfrac{\pi}{2} \right)$ \vphantom{$\left( \dfrac{1\pi}{2} \right)$}

\item $y = -\csc \left( x + \dfrac{\pi}{3} \right)$ \vphantom{$\left( \dfrac{1\pi}{2} \right)$}

\item $y = -\dfrac{1}{3} \sec \left( \dfrac{1}{2}x + \dfrac{\pi}{3} \right)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $y = \csc (2x - \pi)$ \vphantom{$\left( \dfrac{\pi}{2} \right)$}

\item $y = \sec(3x - 2\pi) + 4$ \vphantom{$\left( \dfrac{\pi}{2} \right)$}

\item $y = \csc \left( -x - \dfrac{\pi}{4} \right) - 2$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $y = \cot \left( x + \dfrac{\pi}{6} \right)$ \vphantom{$\left( \dfrac{1\pi}{2} \right)$}

\item $y = -11\cot \left( \dfrac{1}{5} x \right)$

\item $y = \dfrac{1}{3} \cot \left( 2x + \dfrac{3\pi}{2} \right) + 1$ \label{othergraphslast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

In Exercises \ref{expandedsinusoidexerfirst} - \ref{expandedsinusoidexerlast}, use Example \ref{expandedsinusoidex1} as a guide to show that the function is a sinusoid by rewriting it in the forms $C(x) = A \cos(\omega x + \phi) + B$ and $S(x) = A \sin(\omega x + \phi) + B$ for $\omega > 0$ and $0 \leq \phi < 2\pi$.

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = \sqrt{2}\sin(x) + \sqrt{2}\cos(x) + 1$ \label{expandedsinusoidexerfirst}

\item $f(x) = 3\sqrt{3}\sin(3x) - 3\cos(3x)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = -\sin(x) + \cos(x) - 2$ \vphantom{$\left( -\dfrac{1\sqrt{3}}{2} \right)$}

\item $f(x) = -\dfrac{1}{2}\sin(2x) - \dfrac{\sqrt{3}}{2}\cos(2x)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = 2\sqrt{3} \cos(x) - 2\sin(x)$ \vphantom{$\left( -\dfrac{3\sqrt{3}}{2} \right)$}

\item $f(x) = \dfrac{3}{2} \cos(2x) - \dfrac{3\sqrt{3}}{2} \sin(2x) + 6$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = -\dfrac{1}{2} \cos(5x) -\dfrac{\sqrt{3}}{2} \sin(5x)$

\item $f(x) = -6\sqrt{3} \cos(3x) - 6\sin(3x) - 3$ \vphantom{$\left( -\dfrac{\sqrt{3}}{2} \right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = \dfrac{5\sqrt{2}}{2} \sin(x) -\dfrac{5\sqrt{2}}{2} \cos(x)$

\item $f(x) =3 \sin \left(\dfrac{x}{6}\right) -3\sqrt{3} \cos \left(\dfrac{x}{6}\right)$ \vphantom{$\left( \dfrac{-5 \sqrt{3}}{2} \right)$} \label{expandedsinusoidexerlast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item In Exercises \ref{expandedsinusoidexerfirst} - \ref{expandedsinusoidexerlast}, you should have noticed a relationship between the phases $\phi$ for the $S(x)$ and $C(x)$. Show that if $f(x) = A \sin(\omega x + \alpha) + B$, then $f(x) = A \cos(\omega x + \beta) + B$ where $\beta = \alpha - \dfrac{\pi}{2}$.

\label{sinusoidexercise1}

\item Let $\phi$ be an angle measured in radians and let $P(a,b)$ be a point on the terminal side of $\phi$ when it is drawn in standard position. Use Theorem \ref{cosinesinecircle} and the sum identity for sine in Theorem \ref{sinesumdifference} to show that $f(x) = a \, \sin(\omega x) + b\, \cos(\omega x) + B$ (with $\omega > 0$) can be rewritten as $f(x) = \sqrt{a^{2} + b^{2}}\sin(\omega x + \phi) + B$.

\label{sinusoidexercise2}

\item With the help of your classmates, express the domains of the functions in Examples \ref{seccscgraphex} and \ref{tancotgraphex} using extended interval notation. (We will revisit this in Section \ref{TrigEquIneq}.)

\setcounter{HW}{\value{enumi}}

\end{enumerate}

In Exercises \ref{idengraphfirst} - \ref{idengraphlast}, verify the identity by graphing the right and left hand sides on a calculator.

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sin^{2}(x) + \cos^{2}(x) = 1$ \vphantom{$\left( \dfrac{\pi}{2} \right)$} \label{idengraphfirst}

\item $\sec^{2}(x) - \tan^{2}(x) = 1$ \vphantom{$\left( \dfrac{\pi}{2} \right)$}

\item $\cos(x) = \sin\left(\dfrac{\pi}{2} - x\right)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\tan(x+\pi) = \tan(x)$ \vphantom{$\dfrac{\sin(x)}{1+\cos(x)}$}

\item $\sin(2x) = 2\sin(x)\cos(x)$ \vphantom{$\dfrac{\sin(x)}{1+\cos(x)}$}

\item $\tan\left(\dfrac{x}{2}\right) = \dfrac{\sin(x)}{1+\cos(x)}$ \label{idengraphlast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

In Exercises \ref{exploregraphsfirst} - \ref{exploregraphslast}, graph the function with the help of your calculator and discuss the given questions with your classmates.

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = \cos(3x) + \sin(x)$. Is this function periodic? If so, what is the period? \label{exploregraphsfirst}

\item $f(x) = \frac{\sin(x)}{x}$. What appears to be the horizontal asymptote of the graph?

\item $f(x) = x \sin(x)$. Graph $y = \pm x$ on the same set of axes and describe the behavior of $f$.

\item $f(x) = \sin\left(\frac{1}{x}\right)$. What's happening as $x \rightarrow 0$?

\item $f(x) = x - \tan(x)$. Graph $y = x$ on the same set of axes and describe the behavior of $f$.

\item $f(x) = e^{-0.1x} \left( \cos(2x) + \sin(2x)\right)$. Graph $y = \pm e^{-0.1x}$ on the same set of axes and describe the behavior of $f$.

\item $f(x) = e^{-0.1x} \left( \cos(2x) + 2\sin(x)\right)$. Graph $y = \pm e^{-0.1x}$ on the same set of axes and describe the behavior of $f$. \label{exploregraphslast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item Show that a constant function $f$ is periodic by showing that $f(x + 117) = f(x)$ for all real numbers $x$. Then show that $f$ has no period by showing that you cannot find a \emph{smallest} number $p$ such that $f(x + p) = f(x)$ for all real numbers $x$. Said another way, show that $f(x + p) = f(x)$ for all real numbers $x$ for ALL values of $p > 0$, so no smallest value exists to satisfy the definition of `period'.

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\newpage

\subsection{Answers}

\begin{enumerate}

\item \begin{multicols}{2} \raggedcolumns

$y = 3\sin(x)$\\

Period: $2\pi$\\

Amplitude: $3$\\

Phase Shift: $0$\\

Vertical Shift: $0$\\

\begin{mfpic}[25][15]{-0.25}{7}{-3.5}{3.75}

\point[3pt]{(0,0), (1.5708,3), (3.1416, 0), (4.7124,-3), (6.2832,0)}

\axes

\tlabel[cc](7,-0.30){$x$}

\tlabel[cc](0.25,3.75){$y$}

\xmarks{1.5708, 3.1416, 4.7124, 6.2832 }

\ymarks{-3,3}

\tlpointsep{4pt}

\axislabels {x}{{$\frac{\pi}{2}$} 1.5708, {$\pi$} 3.1416, {$\frac{3\pi}{2}$} 4.7124, {$2\pi$} 6.2832}

\axislabels {y}{{$-3$} -3, {$3$} 3}

\function{0, 6.2832, 0.1}{3*sin(x)}

\end{mfpic}

\end{multicols}

\item \begin{multicols}{2} \raggedcolumns

$y = \sin(3x)$\\

Period: $\dfrac{2\pi}{3}$\\

Amplitude: $1$\\

Phase Shift: $0$\\

Vertical Shift: $0$\\

\begin{mfpic}[70][50]{-0.25}{2.5}{-1.25}{1.25}

\point[3pt]{(0,0), (0.5236,1), (1.0472,0), (1.5708,-1), (2.0944,0)}

\axes

\tlabel[cc](2.5,-0.15){$x$}

\tlabel[cc](0.15,1.25){$y$}

\xmarks{0.5236, 1.0472, 1.5708, 2.0944}

\ymarks{-1,1}

\tlpointsep{4pt}

\axislabels {x}{{$\frac{\pi}{6}$} 0.5236, {$\frac{\pi}{3}$} 1.0472, {$\frac{\pi}{2}$} 1.5708, {$\frac{2\pi}{3}$} 2.0944}

\axislabels {y}{{$-1$} -1, {$1$} 1}

\function{0, 2.0944, 0.1}{sin(3*x)}

\end{mfpic}

\end{multicols}

\item \begin{multicols}{2} \raggedcolumns

$y = -2\cos(x)$\\

Period: $2\pi$\\

Amplitude: $2$\\

Phase Shift: $0$\\

Vertical Shift: $0$\\

\begin{mfpic}[25]{-0.25}{7}{-2.5}{2.5}

\point[3pt]{(0,-2), (1.5708,0), (3.1416, 2), (4.7124,0), (6.2832,-2)}

\axes

\tlabel[cc](7,-0.25){$x$}

\tlabel[cc](0.25,2.5){$y$}

\xmarks{1.5708, 3.1416, 4.7124, 6.2832}

\ymarks{-2,2}

\tlpointsep{4pt}

\axislabels {x}{{$\frac{\pi}{2}$} 1.5708, {$\pi$} 3.1416, {$\frac{3\pi}{2}$} 4.7124, {$2\pi$} 6.2832}

\axislabels {y}{{$-2$} -2, {$2$} 2}

\function{0, 6.2832, 0.1}{-2*cos(x)}

\end{mfpic}

\end{multicols}

\item \begin{multicols}{2} \raggedcolumns

$y = \cos \left( x - \dfrac{\pi}{2} \right)$\\

Period: $2\pi$\\

Amplitude: $1$\\

Phase Shift: $\dfrac{\pi}{2}$\\

Vertical Shift: $0$\\

\begin{mfpic}[22][40]{-0.25}{8.3}{-1.5}{1.5}

\point[3pt]{(1.5708,1), (3.1416, 0), (4.7124,-1), (6.2832,0), (7.854,1)}

\axes

\tlabel[cc](8.3,-0.25){$x$}

\tlabel[cc](0.25,1.5){$y$}

\xmarks{1.5708, 3.1416, 4.7124, 6.2832, 7.854}

\ymarks{-1,1}

\tlpointsep{4pt}

\axislabels {x}{{$\frac{\pi}{2}$} 1.5708, {$\pi$} 3.1416, {$\frac{3\pi}{2}$} 4.7124, {$2\pi$} 6.2832, {$\frac{5\pi}{2}$} 7.854}

\axislabels {y}{{$-1$} -1, {$1$} 1}

\function{1.5708, 7.854, 0.1}{cos(x - 1.5708)}

\end{mfpic}

\end{multicols}

\item \begin{multicols}{2} \raggedcolumns

$y = -\sin \left( x + \dfrac{\pi}{3} \right)$\\

Period: $2\pi$\\

Amplitude: $1$\\

Phase Shift: $-\dfrac{\pi}{3}$\\

Vertical Shift: $0$\\

\begin{mfpic}[27][40]{-1.25}{5.75}{-1.5}{1.5}

\point[3pt]{(-1.0472,0), (0.5236,-1), (2.0944,0), (3.6652,1), (5.236,0)}

\axes

\tlabel[cc](5.75,-0.25){$x$}

\tlabel[cc](0.25,1.5){$y$}

\xmarks{-1.0472, 0.5236, 2.0944, 3.6652, 5.236}

\ymarks{-1,1}

\tlpointsep{4pt}

\axislabels {x}{{$-\frac{\pi}{3}$} -1.0472, {$\frac{\pi}{6}$} 0.5236, {$\frac{2\pi}{3}$} 2.0944, {$\frac{7\pi}{6}$} 3.6652, {$\frac{5\pi}{3}$} 5.236}

\axislabels {y}{{$-1$} -1, {$1$} 1}

\function{-1.0472, 5.236, 0.1}{-sin(x + 1.0472)}

\end{mfpic}

\end{multicols}

\item \begin{multicols}{2} \raggedcolumns

$y = \sin(2x - \pi)$\\

Period: $\pi$\\

Amplitude: $1$\\

Phase Shift: $\dfrac{\pi}{2}$\\

Vertical Shift: $0$\\

\begin{mfpic}[35][50]{0}{5.25}{-1.15}{1.5}

\point[3pt]{(1.5708,0), (2.3562,1), (3.1415,0), (3.927,-1), (4.7124,0)}

\axes

\tlabel[cc](5.25,-0.25){$x$}

\tlabel[cc](0.25,1.5){$y$}

\xmarks{1.5708, 2.3562, 3.1415, 3.927, 4.7124}

\ymarks{-1,1}

\tlpointsep{4pt}

\axislabels {x}{{$\frac{\pi}{2}$} 1.5708, {$\frac{3\pi}{4}$} 2.3562, {$\pi$} 3.1415, {$\frac{5\pi}{4}$} 3.927, {$\frac{3\pi}{2}$} 4.7124}

\axislabels {y}{{$-1$} -1, {$1$} 1}

\function{1.5708, 4.7124, 0.1}{sin(2*x - 3.1415)}

\end{mfpic}

\end{multicols}

\item \begin{multicols}{2} \raggedcolumns

$y = -\dfrac{1}{3}\cos \left( \dfrac{1}{2}x + \dfrac{\pi}{3} \right)$\\

Period: $4\pi$\\

Amplitude: $\dfrac{1}{3}$\\

Phase Shift: $-\dfrac{2\pi}{3}$\\

Vertical Shift: $0$\\

\begin{mfpic}[14][100]{-2.25}{11.5}{-0.5}{0.5}

\point[3pt]{(-2.0944, -0.3333), (1.0472, 0), (4.1888, 0.3333), (7.3304, 0), (10.472, -0.3333)}

\axes

\tlabel[cc](11.5,-0.05){$x$}

\tlabel[cc](0.25,0.5){$y$}

\xmarks{-2.0944, 1.0472, 4.1888, 7.3304, 10.472}

\ymarks{-0.3333, 0.3333}

\tlpointsep{4pt}

\axislabels {x}{{$-\frac{2\pi}{3}$} -2.0944, {$\frac{\pi}{3}$} 1.0472, {$\frac{4\pi}{3}$} 4.1888, {$\frac{7\pi}{3}$} 7.3304, {$\frac{10\pi}{3}$} 10.472}

\axislabels {y}{{$-\frac{1}{3}$} -0.3333, {$\frac{1}{3}$} 0.3333}

\function{-2.0944, 10.472, 0.1}{-0.3333*cos(0.5*x + 1.0472)}

\end{mfpic}

\end{multicols}

\item \begin{multicols}{2} \raggedcolumns

$y = \cos (3x - 2\pi) + 4$\\

Period: $\dfrac{2\pi}{3}$\\

Amplitude: $1$\\

Phase Shift: $\dfrac{2\pi}{3}$\\

Vertical Shift: 4\\

\begin{mfpic}[36][25]{-0.5}{5}{-0.5}{5.5}

\point[3pt]{(2.0944,5), (2.618,4), (3.1415,3), (3.6652,4), (4.1888,5)}

\axes

\tlabel[cc](5,-0.25){$x$}

\tlabel[cc](0.25,5.5){$y$}

\xmarks{2.0944, 2.618, 3.1415, 3.6652, 4.1888}

\ymarks{3,4,5}

\tlpointsep{4pt}

\axislabels {x}{{$\frac{2\pi}{3}$} 2.0944, {$\frac{5\pi}{6}$} 2.618, {$\pi$} 3.1415, {$\frac{7\pi}{6}$} 3.6652, {$\frac{4\pi}{3}$} 4.1888}

\axislabels {y}{{$3$} 3, {$4$} 4, {$5$} 5}

\function{2.0944, 4.1888, 0.1}{cos(3*x - 6.2834) + 4}

\end{mfpic}

\end{multicols}

\item \begin{multicols}{2} \raggedcolumns

$y = \sin \left( -x - \dfrac{\pi}{4} \right) - 2$ \\

Period: $2\pi$\\

Amplitude: $1$\\

Phase Shift: $-\dfrac{\pi}{4}$ (You need to use \\ \vspace*{.1in}

$y = -\sin \left( x + \dfrac{\pi}{4} \right) - 2 $ to find this.)\footnote{Two cycles of the graph are shown to illustrate the discrepancy discussed on page \pageref{phaseshiftissue}.}\\

Vertical Shift: $-2$\\

\begin{mfpic}[13][27]{-7.5}{6.5}{-3.25}{0.5}

\point[3pt]{(-7.0686,-2), (-5.4979,-3), (-3.927,-2), (-2.3562,-1), (-0.7854,-2), (0.7854,-3), (2.3562,-2), (3.927,-1), (5.4979,-2)}

\axes

\tlabel[cc](6.5,-0.25){$x$}

\tlabel[cc](0.25,0.5){$y$}

\xmarks{-7.0686, -5.4979, -3.927, -2.3562,-0.7854, 0.7854, 2.3562, 3.927, 5.4979}

\ymarks{-3,-2,-1}

\tlpointsep{5pt}

\axislabels {x}{{$-\frac{9\pi}{4} \hspace{6pt}$} -7.0686, {$-\frac{7\pi}{4} \hspace{6pt}$} -5.4979, {$-\frac{5\pi}{4} \hspace{6pt}$} -3.927, {$-\frac{3\pi}{4} \hspace{6pt}$} -2.3562, {$-\frac{\pi}{4} \hspace{6pt}$} -0.7854, {$\frac{\pi}{4}$} 0.7854, {$\frac{3\pi}{4}$} 2.3562, {$\frac{5\pi}{4}$} 3.927, {$\frac{7\pi}{4}$} 5.4979}

\axislabels {y}{{$-3$} -3, {$-2$} -2, {$-1$} -1}

\function{-7.0686, 5.4979, 0.1}{-1*sin(x + 0.7854) - 2}

\end{mfpic}

\end{multicols}

\item \begin{multicols}{2} \raggedcolumns

$y = \dfrac{2}{3} \cos \left( \dfrac{\pi}{2} - 4x \right) + 1$\\

Period: $\dfrac{\pi}{2}$\\

Amplitude: $\dfrac{2}{3}$\\

Phase Shift: $\dfrac{\pi}{8}$ (You need to use \\

$y = \dfrac{2}{3} \cos \left( 4x - \dfrac{\pi}{2} \right) + 1$ to find this.)\footnote{Again, we graph two cycles to illustrate the discrepancy discussed on page \pageref{phaseshiftissue}.}\\

Vertical Shift: $1$\\

\begin{mfpic}[52][45]{-1.5}{2.25}{-0.25}{2}

\point[3pt]{(-1.1781, 1.6667), (-0.7854, 1), (-0.3927, 0.3333), (0, 1), (0.3927, 1.6667), (0.7854, 1), (1.1781, 0.3333), (1.5708, 1), (1.9635, 1.6667)}

\axes

\tlabel[cc](2.25,-0.25){$x$}

\tlabel[cc](0.15,2){$y$}

\xmarks{-1.1781, -0.7854, -0.3927, 0.3927, 0.7854, 1.1781, 1.5708, 1.9635}

\ymarks{0.3333, 1, 1.6667}

\tlpointsep{4pt}

\axislabels {x}{{$-\frac{3\pi}{8} \hspace{6pt}$} -1.1781, {$-\frac{\pi}{4} \hspace{6pt}$} -0.7854, {$-\frac{\pi}{8} \hspace{6pt}$} -0.3927, {$\frac{\pi}{8}$} 0.3927, {$\frac{\pi}{4}$} 0.7854, {$\frac{3\pi}{8}$} 1.1781, {$\frac{\pi}{2}$} 1.5708, {$\frac{5\pi}{8}$} 1.9635}

\axislabels {y}{{$\frac{1}{3}$} 0.333, {$1$} 1, {$\frac{5}{3}$} 1.6667}

\function{-1.1781, 1.9635, 0.1}{0.6667*cos(4*x - 1.5708) + 1}

\end{mfpic}

\end{multicols}

\item \begin{multicols}{2} \raggedcolumns

$y = -\dfrac{3}{2} \cos \left( 2x + \dfrac{\pi}{3} \right) - \dfrac{1}{2}$\\

Period: $\pi$\\

Amplitude: $\dfrac{3}{2}$\\

Phase Shift: $-\dfrac{\pi}{6}$\\

Vertical Shift: $-\dfrac{1}{2}$\\

\begin{mfpic}[51][30]{-.75}{3}{-2.25}{1.5}

\point[3pt]{(-0.5236,-2), (0.2618,-0.5), (1.0472, 1), (1.8326, -0.5), (2.618, -2)}

\axes

\tlabel[cc](3,-0.25){$x$}

\tlabel[cc](0.15,1.5){$y$}

\xmarks{-0.5236, 0.2618, 1.0472, 1.8326, 2.618}

\ymarks{-2, -0.5, 1}

\tlpointsep{4pt}

\axislabels {x}{{$-\frac{\pi}{6}$} -0.5236, {$\frac{\pi}{12}$} 0.2618, {$\frac{\pi}{3}$} 1.0472, {$\frac{7\pi}{12}$} 1.8326, {$\frac{5\pi}{6}$} 2.618}

\axislabels {y}{{$-2$} -2, {$-\frac{1}{2}$} -0.5, {$1$} 1}

\function{-0.5236, 2.618, 0.1}{-1.5*cos(2*x + 1.047) - 0.5}

\end{mfpic}

\end{multicols}

\item \begin{multicols}{2} \raggedcolumns

$y = 4\sin (-2\pi x + \pi)$ \\

Period: $1$\\

Amplitude: $4$\\

Phase Shift: $\dfrac{1}{2}$ (You need to use \\

$y = -4\sin (2\pi x - \pi)$ to find this.)\footnote{This will be the last time we graph two cycles to illustrate the discrepancy discussed on page \pageref{phaseshiftissue}.}\\

Vertical Shift: $0$\\

\begin{mfpic}[80][12]{-.75}{1.75}{-4.5}{4.75}

\point[3pt]{(-0.5,0), (-0.25,-4), (0,0), (0.25,4), (0.5,0), (0.75,-4), (1,0), (1.25,4), (1.5,0)}

\axes

\tlabel[cc](1.75,-0.5){$x$}

\tlabel[cc](0.1,4.75){$y$}

\xmarks{-0.5, -0.25, 0.25, 0.5, 0.75, 1, 1.25, 1.5}

\ymarks{-4,4}

\tlpointsep{4pt}

\axislabels {x}{{$-\frac{1}{2}$ \hspace{6pt}} -0.5, {$-\frac{1}{4}$ \hspace{6pt}} -0.25, {$\frac{1}{4}$} 0.25, {$\frac{1}{2}$} 0.5, {$\frac{3}{4}$} 0.75, {$1$} 1, {$\frac{5}{4}$} 1.25, {$\frac{3}{2}$} 1.5}

\axislabels {y}{{$-4$} -4, {$4$} 4}

\function{-0.5, 1.5, 0.01}{(-4)*sin((6.2831853*x) - 3.14159265)}

\end{mfpic}

\end{multicols}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \begin{multicols}{2} \raggedcolumns

$y = \tan \left(x - \dfrac{\pi}{3} \right)$\\

Period: $\pi$\\

\begin{mfpic}[46][18]{-1}{3}{-5}{5}

\point[3pt]{(0.2618,-1), (1.0472,0), (1.8326,1)}

\axes

\tlabel[cc](3,-0.5){$x$}

\tlabel[cc](0.25,5){$y$}

\xmarks{0.2618, 1.0472, 1.8326}

\ymarks{-1,1}

\tlpointsep{4pt}

\axislabels {x}{{$-\frac{\pi}{6}$ \hspace{11pt}} -0.5236, {$\frac{\pi}{12}$} 0.2618, {$\frac{\pi}{3}$} 1.0472, {$\frac{7\pi}{12}$} 1.8326, {$\frac{5\pi}{6}$ \hspace{11pt}} 2.618}

\axislabels {y}{{$-1$} -1, {$1$} 1}

\arrow \reverse \arrow \function{-0.30, 2.40, 0.1}{tan(x - 1.0472)}

\dashed \polyline{(-0.5236,-5), (-0.5236,5)}

\dashed \polyline{(2.618,-5),(2.618,5)}

\end{mfpic}

\end{multicols}

\item \begin{multicols}{2} \raggedcolumns

$y = 2\tan \left( \dfrac{1}{4}x \right) - 3$\\

Period: $4\pi$

\begin{mfpic}[13][13]{-7}{8}{-10}{4}

\point[3pt]{(-3.1416,-5), (0,-3), (3.1416,-1)}

\axes

\tlabel[cc](8,-0.5){$x$}

\tlabel[cc](0.5,4){$y$}

\xmarks{-3.1416, 3.1416}

\ymarks{-5, -3, -1}

\tlpointsep{4pt}

\axislabels {x}{{$-2\pi$} -6.2832, {$-\pi$ \hspace{6pt}} -3.1416, {$\pi$} 3.1416, {\hspace{11pt}$2\pi$} 6.2832}

\axislabels {y}{{$-5$} -5, {$-3$} -3, {$-1$} -1}

\arrow \reverse \arrow \function{-5.1, 5.1, 0.1}{2*tan(0.25*x) - 3}

\dashed \polyline{(-6.2832,-10), (-6.2832,4)}

\dashed \polyline{(6.2382,-10),(6.2832,4)}

\end{mfpic}

\end{multicols}

\item \begin{multicols}{2} \raggedcolumns

$y = \dfrac{1}{3}\tan(-2x - \pi) + 1$ \\

is equivalent to \\

$y = -\dfrac{1}{3}\tan(2x + \pi) + 1$ \\

via the Even / Odd identity for tangent.\\

Period: $\dfrac{\pi}{2}$\\

\begin{mfpic}[54][36]{-3}{0.5}{-2}{2.5}

\point[3pt]{(-1.9635,1.3333),(-1.5708,1),(-1.1781,0.6667)}

\axes

\tlabel[cc](0.5,-0.25){$x$}

\tlabel[cc](0.25,2.5){$y$}

\xmarks{-1.9635,-1.5708,-1.1781}

\ymarks{0.6667,1,1.3333}

\tlpointsep{4pt}

\small

\axislabels {x}{{$-\frac{3\pi}{4}$ \hspace{11pt}} -2.3562, {$-\frac{5\pi}{8}$ \hspace{6pt}} -1.9635, {$-\frac{\pi}{2}$ \hspace{6pt}} -1.5708, {$-\frac{3\pi}{8}$ \hspace{6pt}} -1.1781, {$-\frac{\pi}{4}$} -0.7854}

\axislabels {y}{{$\frac{4}{3}$} 1.3333, {$1$} 1, {$\frac{2}{3}$} 0.6667}

\normalsize

\arrow \reverse \arrow \function{-2.25, -0.84, 0.1}{0.3333*tan(-2*x - 3.1416) + 1}

\dashed \polyline{(-2.3562,-2), (-2.3562,2.5)}

\dashed \polyline{(-0.7854,-2),(-0.7854,2.5)}

\end{mfpic}

\end{multicols}

\item \begin{multicols}{2} \raggedcolumns

$y = \sec \left( x - \frac{\pi}{2} \right)$ \\

Start with $y = \cos \left( x - \frac{\pi}{2} \right)$\\

Period: $2\pi$\\

\begin{mfpic}[22][20]{-0.25}{8.3}{-4}{4}

\point[3pt]{(1.5708,1), (4.7124,-1), (7.854,1)}

\axes

\tlabel[cc](8.3,-0.25){$x$}

\tlabel[cc](0.25,4){$y$}

\xmarks{1.5708, 3.1416, 4.7124, 6.2832, 7.854}

\ymarks{-1,1}

\tlpointsep{4pt}

\axislabels {x}{{$\frac{\pi}{2}$} 1.5708, {$\pi$} 3.1416, {$\frac{3\pi}{2}$} 4.7124, {$2\pi$} 6.2832, {$\frac{5\pi}{2}$} 7.854}

\axislabels {y}{{$-1$} -1, {$1$} 1}

\dashed \polyline{(6.2832,-4),(6.2832,4)}

\dashed \polyline{(3.1416,-4),(3.1416,4)}

\dotted[1pt, 3pt] \function{1.5708, 7.854, 0.1}{cos(x - 1.5708)}

\arrow \reverse \function{6.55, 7.854, 0.1}{1/(cos(x - 1.5708))}

\arrow \reverse \arrow \function{3.4084, 6.0164, 0.1}{1/(cos(x - 1.5708))}

\arrow \function{1.5708, 2.8748, 0.1}{1/(cos(x - 1.5708))}

\end{mfpic}

\end{multicols}

\item \begin{multicols}{2} \raggedcolumns

$y = -\csc \left( x + \dfrac{\pi}{3} \right)$\\

Start with $y = -\sin \left( x + \dfrac{\pi}{3} \right)$\\

Period: $2\pi$

\begin{mfpic}[27][20]{-1.25}{5.75}{-4}{4}

\point[3pt]{(0.5236,-1), (3.6652,1)}

\axes

\tlabel[cc](5.75,-0.25){$x$}

\tlabel[cc](0.25,4){$y$}

\xmarks{-1.0472, 0.5236, 2.0944, 3.6652, 5.236}

\ymarks{-1,1}

\tlpointsep{4pt}

\axislabels {x}{{$-\frac{\pi}{3}$} -1.0472, {$\frac{\pi}{6}$} 0.5236, {$\frac{2\pi}{3}$} 2.0944, {$\frac{7\pi}{6}$} 3.6652, {$\frac{5\pi}{3}$} 5.236}

\axislabels {y}{{$-1$} -1, {$1$} 1}

\dashed \polyline{(-1.0472,-4),(-1.0472,4)}

\dashed \polyline{(2.0944,-4),(2.0944,4)}

\dashed \polyline{(5.236,-4),(5.236,4)}

\dotted[1pt, 3pt] \function{-1.0472, 5.236, 0.1}{-sin(x + 1.0472)}

\arrow \reverse \arrow \function{-0.794, 1.841, 0.1}{-1/(sin(x + 1.0472))}

\arrow \reverse \arrow \function{2.347, 4.98, 0.1}{-1/(sin(x + 1.0472))}

\end{mfpic}

\end{multicols}

\item \begin{multicols}{2} \raggedcolumns

$y = -\dfrac{1}{3} \sec \left( \dfrac{1}{2}x + \dfrac{\pi}{3} \right)$\\

Start with $y = -\dfrac{1}{3}\cos \left( \dfrac{1}{2}x + \dfrac{\pi}{3} \right)$\\

Period: $4\pi$

\begin{mfpic}[14][70]{-2.25}{11.1}{-1.5}{1.5}

\point[3pt]{(-2.0944, -0.3333), (4.1888, 0.3333), (10.472, -0.3333)}

\axes

\tlabel[cc](11.3,-0.1){$x$}

\tlabel[cc](0.25,1.5){$y$}

\xmarks{-2.0944, 1.0472, 4.1888, 7.3304, 10.472}

\ymarks{-0.3333, 0.3333}

\tlpointsep{4pt}

\axislabels {x}{{$-\frac{2\pi}{3}$} -2.0944, {$\frac{\pi}{3}$} 1.0472, {$\frac{4\pi}{3}$} 4.1888, {$\frac{7\pi}{3}$} 7.3304, {$\frac{10\pi}{3}$} 10.472}

\axislabels {y}{{$-\frac{1}{3}$} -0.3333, {$\frac{1}{3}$} 0.3333}

\dotted[1pt, 3pt] \function{-2.0944, 10.472, 0.1}{-0.3333*cos(0.5*x + 1.0472)}

\dashed \polyline{(1.0472,-1.5),(1.0472,1.5)}

\dashed \polyline{(7.3304,-1.5),(7.3304,1.5)}

\arrow \function{-2.0944, 0.6, 0.1}{-0.3333/(cos(0.5*x + 1.0472))}

\arrow \reverse \arrow \function{1.4944, 6.8832, 0.1}{-0.3333/(cos(0.5*x + 1.0472))}

\arrow \reverse \function{7.777, 10.472, 0.1}{-0.3333/(cos(0.5*x + 1.0472))}

\end{mfpic}

\end{multicols}

\item \begin{multicols}{2} \raggedcolumns

$y = \csc (2x - \pi)$\\

Start with $y = \sin(2x - \pi)$\\

Period: $\pi$\\

\begin{mfpic}[36][22]{0}{5.15}{-4}{4}

\point[3pt]{(2.3562,1), (3.927,-1)}

\axes

\tlabel[cc](5.15,-0.25){$x$}

\tlabel[cc](0.25,4){$y$}

\xmarks{1.5708, 2.3562, 3.1415, 3.927, 4.7124}

\ymarks{-1,1}

\tlpointsep{4pt}

\axislabels {x}{{$\frac{\pi}{2}$} 1.5708, {$\frac{3\pi}{4}$} 2.3562, {$\pi$} 3.1415, {$\frac{5\pi}{4}$} 3.927, {$\frac{3\pi}{2}$} 4.7124}

\axislabels {y}{{$-1$} -1, {$1$} 1}

\dotted[1pt, 3pt] \function{1.5708, 4.7124, 0.1}{sin(2*x - 3.1415)}

\dashed \polyline{(1.5708,-4),(1.5708,4)}

\dashed \polyline{(3.1415,-4),(3.1415,4)}

\dashed \polyline{(4.7124,-4),(4.7124,4)}

\arrow \reverse \arrow \function{1.6973, 3.015, 0.1}{1/(sin(2*x - 3.1415))}

\arrow \reverse \arrow \function{3.268, 4.5859, 0.1}{1/(sin(2*x - 3.1415))}

\end{mfpic}

\end{multicols}

\item \begin{multicols}{2} \raggedcolumns

$y = \sec(3x - 2\pi) + 4$\\

Start with $y = \cos (3x - 2\pi) + 4$\\

Period: $\dfrac{2\pi}{3}$\\

\begin{mfpic}[35][19]{-1}{4.73}{-0.5}{8}

\point[3pt]{(2.0944,5), (3.1415,3), (4.1888,5)}

\axes

\tlabel[cc](4.73,-0.25){$x$}

\tlabel[cc](0.25,8){$y$}

\xmarks{2.0944, 2.618, 3.1415, 3.6652, 4.1888}

\ymarks{3,4,5}

\tlpointsep{4pt}

\axislabels {x}{{$\frac{2\pi}{3}$} 2.0944, {$\frac{5\pi}{6}$} 2.618, {$\pi$} 3.1415, {$\frac{7\pi}{6}$} 3.6652, {$\frac{4\pi}{3}$} 4.1888}

\axislabels {y}{{$3$} 3, {$4$} 4, {$5$} 5}

\dotted[1pt, 3pt] \function{2.0944, 4.1888, 0.1}{cos(3*x - 6.2834) + 4}

\dashed \polyline {(2.618,-1),(2.618,8)}

\dashed \polyline {(3.6652,-1),(3.6652,8)}

\arrow \function{2.0944, 2.533, 0.1}{1/(cos(3*x - 6.2834)) + 4}

\arrow \reverse \arrow \function{2.69, 3.593, 0.1}{1/(cos(3*x - 6.2834)) + 4}

\arrow \reverse \function{3.7502, 4.1888, 0.1}{1/(cos(3*x - 6.2834)) + 4}

\end{mfpic}

\end{multicols}

\item \begin{multicols}{2} \raggedcolumns

$y = \csc \left( -x - \dfrac{\pi}{4} \right) - 2$\\

Start with $y = \sin \left( -x - \dfrac{\pi}{4} \right) - 2$ \\

Period: $2\pi$\\

\begin{mfpic}[28][22]{-1}{6}{-6}{2}

\point[3pt]{(0.7854,-3), (3.927,-1)}

\axes

\tlabel[cc](6,-0.25){$x$}

\tlabel[cc](0.25,2){$y$}

\xmarks{-0.7854, 0.7854, 2.3562, 3.927, 5.4979}

\ymarks{-3,-2,-1}

\tlpointsep{4pt}

\axislabels {x}{{$-\frac{\pi}{4}$} -0.7854, {$\frac{\pi}{4}$} 0.7854, {$\frac{3\pi}{4}$} 2.3562, {$\frac{5\pi}{4}$} 3.927, {$\frac{7\pi}{4}$} 5.4979}

\axislabels {y}{{$-3$} -3, {$-2$} -2, {$-1$} -1}

\dotted[1pt, 3pt] \function{-0.7854, 5.4979, 0.1}{-1*sin(x + 0.7854) - 2}

\dashed \polyline{(-0.7854,-6),(-0.7854,2)}

\dashed \polyline{(2.3562,-6),(2.3562,2)}

\dashed \polyline{(5.4979,-6),(5.4979,2)}

\arrow \reverse \arrow \function{-0.5324, 2.1032, 0.1}{-1/(sin(x + 0.7854)) - 2}

\arrow \reverse \arrow \function{2.6092, 5.2449, 0.1}{-1/(sin(x + 0.7854)) - 2}

\end{mfpic}

\end{multicols}

\item \begin{multicols}{2} \raggedcolumns

$y = \cot \left( x + \dfrac{\pi}{6} \right)$\\

Period: $\pi$\\

\begin{mfpic}[50][24]{-.75}{3}{-4}{4}

\point[3pt]{(0.2618,1), (1.0472, 0), (1.8326, -1)}

\axes

\tlabel[cc](3,-0.25){$x$}

\tlabel[cc](0.15,4){$y$}

\xmarks{-0.5236, 0.2618, 1.0472, 1.8326, 2.618}

\ymarks{-1, 1}

\tlpointsep{4pt}

\axislabels {x}{{$-\frac{\pi}{6}$} -0.5236, {$\frac{\pi}{12}$} 0.2618, {$\frac{\pi}{3}$} 1.0472, {$\frac{7\pi}{12}$} 1.8326, {$\frac{5\pi}{6}$} 2.618}

\axislabels {y}{{$-1$} -1, {$1$} 1}

\arrow \reverse \arrow \function{-0.278, 2.37, 0.1}{cot(x + 0.5236)}

\dashed \polyline{(-0.5236,-4),(-0.5236,4)}

\dashed \polyline{(2.618,-4),(2.618,4)}

\end{mfpic}

\end{multicols}

\item \begin{multicols}{2} \raggedcolumns

$y = -11\cot \left( \dfrac{1}{5} x \right)$\\

Period: $5\pi$\\

\begin{mfpic}[20][20]{-1}{8}{-4}{4}

\point[3pt]{(1.5708,-1), (3.1416, 0), (4.7124,1)}

\axes

\tlabel[cc](8,-0.5){$x$}

\tlabel[cc](0.5,4){$y$}

\xmarks{1.5708, 3.1416, 4.7124, 6.2832}

\ymarks{-1,1}

\tlpointsep{4pt}

\axislabels {x}{{$\frac{5\pi}{4}$} 1.5708, {$\frac{5\pi}{2}$} 3.1416, {$\frac{15\pi}{4}$} 4.7124, {$5\pi$} 6.2832}

\axislabels {y}{{$-11$} -1, {$11$} 1}

\arrow \reverse \arrow \function{0.5, 5.8, 0.1}{-1*cot(x/2)}

\dashed \polyline{(6.2832,-4), (6.2832,4)}

\end{mfpic}

\end{multicols}

\item \begin{multicols}{2} \raggedcolumns

$y = \dfrac{1}{3} \cot \left( 2x + \dfrac{3\pi}{2} \right) + 1$\\

Period: $\dfrac{\pi}{2}$

\begin{mfpic}[50][40]{-3}{0.5}{-2}{2.5}

\point[3pt]{(-1.9635,1.3333),(-1.5708,1),(-1.1781,0.6667)}

\axes

\tlabel[cc](0.5,-0.25){$x$}

\tlabel[cc](0.25,2.5){$y$}

\xmarks{-1.9635,-1.5708,-1.1781}

\ymarks{0.6667,1,1.3333}

\tlpointsep{4pt}

\small

\axislabels {x}{{$-\frac{3\pi}{4}$ \hspace{11pt}} -2.3562, {$-\frac{5\pi}{8}$ \hspace{6pt}} -1.9635, {$-\frac{\pi}{2}$ \hspace{6pt}} -1.5708, {$-\frac{3\pi}{8}$ \hspace{6pt}} -1.1781, {$-\frac{\pi}{4}$} -0.7854}

\axislabels {y}{{$\frac{4}{3}$} 1.3333, {$1$} 1, {$\frac{2}{3}$} 0.6667}

\normalsize

\arrow \reverse \arrow \function{-2.25, -0.84, 0.1}{0.3333*tan(-2*x - 3.1416) + 1}

\dashed \polyline{(-2.3562,-2), (-2.3562,2.5)}

\dashed \polyline{(-0.7854,-2),(-0.7854,2.5)}

\end{mfpic}

\end{multicols}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = \sqrt{2}\sin(x) + \sqrt{2}\cos(x) + 1 = 2\sin\left(x + \dfrac{\pi}{4}\right) + 1 = 2\cos\left(x + \dfrac{7\pi}{4}\right) + 1$

\item $f(x) = 3\sqrt{3}\sin(3x) - 3\cos(3x) = 6\sin\left(3x + \dfrac{11\pi}{6}\right) = 6\cos\left(3x + \dfrac{4\pi}{3}\right)$

\item $f(x) = -\sin(x) + \cos(x) - 2 = \sqrt{2}\sin\left(x + \dfrac{3\pi}{4}\right) - 2 = \sqrt{2}\cos\left(x + \dfrac{\pi}{4}\right) - 2$

\item $f(x) = -\dfrac{1}{2}\sin(2x) - \dfrac{\sqrt{3}}{2}\cos(2x) = \sin\left(2x + \dfrac{4\pi}{3}\right) = \cos\left(2x + \dfrac{5\pi}{6}\right)$

\item $f(x) = 2\sqrt{3} \cos(x) - 2\sin(x) = 4\sin\left(x+\dfrac{2\pi}{3} \right) = 4\cos\left(x + \dfrac{\pi}{6}\right)$

\item $f(x) = \dfrac{3}{2} \cos(2x) - \dfrac{3\sqrt{3}}{2} \sin(2x) + 6 =3\sin\left(2x + \dfrac{5\pi}{6}\right) + 6 = 3\cos\left(2x + \dfrac{\pi}{3}\right) + 6$

\item $f(x) = -\dfrac{1}{2} \cos(5x) -\dfrac{\sqrt{3}}{2} \sin(5x) = \sin\left(5x + \dfrac{7\pi}{6}\right) = \cos\left(5x + \dfrac{2\pi}{3}\right)$

\item $f(x) = -6\sqrt{3} \cos(3x) - 6\sin(3x) - 3 = 12\sin\left(3x + \dfrac{4\pi}{3}\right) - 3 = 12\cos\left(3x + \dfrac{5\pi}{6}\right) - 3$

\item $f(x) = \dfrac{5\sqrt{2}}{2} \sin(x) -\dfrac{5\sqrt{2}}{2} \cos(x) = 5\sin\left(x + \dfrac{7\pi}{4}\right)= 5\cos\left(x + \dfrac{5\pi}{4}\right)$

\item $f(x) =3\sin\left(\dfrac{x}{6}\right) -3\sqrt{3} \cos\left(\dfrac{x}{6}\right) = 6\sin\left( \dfrac{x}{6}+\dfrac{5\pi}{3}\right)= 6\cos\left( \dfrac{x}{6}+\dfrac{7\pi}{6}\right) $

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\closegraphsfile

10.6: The Inverse Trigonometric Functions

\subsection{Exercises}

In Exercises \ref{exactvaluearcfirst} - \ref{exactvaluearclast}, find the exact value.

\begin{multicols}{4}

\begin{enumerate}

\item $\arcsin \left( -1 \right)$ \vphantom{$\left( -\dfrac{\sqrt{3}}{2} \right)$} \label{exactvaluearcfirst}

\item $\arcsin \left( -\dfrac{\sqrt{3}}{2} \right)$

\item $\arcsin \left( -\dfrac{\sqrt{2}}{2} \right)$

\item $\arcsin \left( -\dfrac{1}{2} \right)$ \vphantom{$\left( -\dfrac{\sqrt{3}}{2} \right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{4}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\arcsin \left( 0 \right)$ \vphantom{$\left( \dfrac{\sqrt{3}}{2} \right)$}

\item $\arcsin \left( \dfrac{1}{2} \right)$ \vphantom{$\left( \dfrac{\sqrt{3}}{2} \right)$}

\item $\arcsin \left( \dfrac{\sqrt{2}}{2} \right)$

\item $\arcsin \left( \dfrac{\sqrt{3}}{2} \right)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{4}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\arcsin \left( 1 \right)$ \vphantom{$\left( -\dfrac{\sqrt{3}}{2} \right)$}

\item $\arccos \left( -1 \right)$ \vphantom{$\left( -\dfrac{\sqrt{3}}{2} \right)$}

\item $\arccos \left( -\dfrac{\sqrt{3}}{2} \right)$

\item $\arccos \left( -\dfrac{\sqrt{2}}{2} \right)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{4}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\arccos \left( -\dfrac{1}{2} \right)$ \vphantom{$\left( \dfrac{\sqrt{3}}{2} \right)$}

\item $\arccos \left( 0 \right)$ \vphantom{$\left( \dfrac{\sqrt{3}}{2} \right)$}

\item $\arccos \left( \dfrac{1}{2} \right)$ \vphantom{$\left( \dfrac{\sqrt{3}}{2} \right)$}

\item $\arccos \left( \dfrac{\sqrt{2}}{2} \right)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{4}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\arccos \left( \dfrac{\sqrt{3}}{2} \right)$

\item $\arccos \left( 1 \right)$ \vphantom{$\left( \dfrac{\sqrt{3}}{2} \right)$}

\item $\arctan \left( -\sqrt{3} \right)$ \vphantom{$\left( \dfrac{\sqrt{3}}{2} \right)$}

\item $\arctan \left( -1 \right)$ \vphantom{$\left( \dfrac{\sqrt{3}}{2} \right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{4}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\arctan \left( -\dfrac{\sqrt{3}}{3} \right)$

\item $\arctan \left( 0 \right)$ \vphantom{$\left( -\dfrac{\sqrt{3}}{2} \right)$}

\item $\arctan \left( \dfrac{\sqrt{3}}{3} \right)$

\item $\arctan \left( 1 \right)$ \vphantom{$\left( -\dfrac{\sqrt{3}}{2} \right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{4}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\arctan \left( \sqrt{3} \right)$ \vphantom{$\left( -\dfrac{\sqrt{3}}{2} \right)$}

\item $\mbox{arccot} \left( -\sqrt{3} \right)$ \vphantom{$\left( -\dfrac{\sqrt{3}}{2} \right)$}

\item $\mbox{arccot} \left( -1 \right)$ \vphantom{$\left( -\dfrac{\sqrt{3}}{2} \right)$}

\item $\mbox{arccot} \left( -\dfrac{\sqrt{3}}{3} \right)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{4}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\mbox{arccot} \left( 0 \right)$ \vphantom{$\left( -\dfrac{\sqrt{3}}{2} \right)$}

\item $\mbox{arccot} \left( \dfrac{\sqrt{3}}{3} \right)$

\item $\mbox{arccot} \left( 1 \right)$ \vphantom{$\left( -\dfrac{\sqrt{3}}{2} \right)$}

\item $\mbox{arccot} \left( \sqrt{3} \right)$ \vphantom{$\left( -\dfrac{\sqrt{3}}{2} \right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{4}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\mbox{arcsec} \left( 2 \right)$

\item $\mbox{arccsc} \left( 2 \right)$

\item $\mbox{arcsec} \left( \sqrt{2} \right)$

\item $\mbox{arccsc} \left( \sqrt{2} \right)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{4}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\mbox{arcsec} \left( \dfrac{2\sqrt{3}}{3} \right)$

\item $\mbox{arccsc} \left( \dfrac{2\sqrt{3}}{3} \right)$

\item $\mbox{arcsec} \left( 1 \right)$ \vphantom{$\left( -\dfrac{\sqrt{3}}{2} \right)$}

\item $\mbox{arccsc} \left( 1 \right)$ \vphantom{$\left( -\dfrac{\sqrt{3}}{2} \right)$} \label{exactvaluearclast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

In Exercises \ref{calcfriendexactfirst} - \ref{calcfriendexactlast}, assume that the range of arcsecant is $\left[0, \frac{\pi}{2} \right) \cup \left[\pi, \frac{3\pi}{2} \right)$ and that the range of arccosecant is $\left(0, \frac{\pi}{2} \right] \cup \left( \pi, \frac{3\pi}{2} \right]$ when finding the exact value.

\begin{multicols}{4}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\mbox{arcsec} \left( -2 \right)$ \vphantom{$\left( -\dfrac{2\sqrt{3}}{3} \right)$} \label{calcfriendexactfirst}

\item $\mbox{arcsec} \left( -\sqrt{2} \right)$ \vphantom{$\left( -\dfrac{2\sqrt{3}}{3} \right)$}

\item $\mbox{arcsec} \left( -\dfrac{2\sqrt{3}}{3} \right)$

\item $\mbox{arcsec} \left( -1 \right)$ \vphantom{$\left( -\dfrac{2\sqrt{3}}{3} \right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{4}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\mbox{arccsc} \left( -2 \right)$ \vphantom{$\left( -\dfrac{2\sqrt{3}}{3} \right)$}

\item $\mbox{arccsc} \left( -\sqrt{2} \right)$ \vphantom{$\left( -\dfrac{2\sqrt{3}}{3} \right)$}

\item $\mbox{arccsc} \left( -\dfrac{2\sqrt{3}}{3} \right)$

\item $\mbox{arccsc} \left( -1 \right)$ \vphantom{$\left( -\dfrac{2\sqrt{3}}{3} \right)$} \label{calcfriendexactlast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\pagebreak

In Exercises \ref{trigfriendexactfirst} - \ref{trigfriendexactlast}, assume that the range of arcsecant is $\left[0, \frac{\pi}{2} \right) \cup \left( \frac{\pi}{2}, \pi \right]$ and that the range of arccosecant is

$\left[ -\frac{\pi}{2}, 0 \right) \cup \left(0, \frac{\pi}{2} \right]$ when finding the exact value.

\begin{multicols}{4}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\mbox{arcsec} \left( -2 \right)$ \vphantom{$\left( -\dfrac{2\sqrt{3}}{3} \right)$} \label{trigfriendexactfirst}

\item $\mbox{arcsec} \left( -\sqrt{2} \right)$ \vphantom{$\left( -\dfrac{2\sqrt{3}}{3} \right)$}

\item $\mbox{arcsec} \left( -\dfrac{2\sqrt{3}}{3} \right)$

\item $\mbox{arcsec} \left( -1 \right)$ \vphantom{$\left( -\dfrac{2\sqrt{3}}{3} \right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{4}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\mbox{arccsc} \left( -2 \right)$ \vphantom{$\left( -\dfrac{2\sqrt{3}}{3} \right)$}

\item $\mbox{arccsc} \left( -\sqrt{2} \right)$ \vphantom{$\left( -\dfrac{2\sqrt{3}}{3} \right)$}

\item $\mbox{arccsc} \left( -\dfrac{2\sqrt{3}}{3} \right)$

\item $\mbox{arccsc} \left( -1 \right)$ \vphantom{$\left( -\dfrac{2\sqrt{3}}{3} \right)$} \label{trigfriendexactlast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

In Exercises \ref{comboexactfirst} - \ref{comboexactlast}, find the exact value or state that it is undefined.

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sin\left(\arcsin\left(\dfrac{1}{2}\right)\right)$ \vphantom{$\left( -\dfrac{\sqrt{2}}{2} \right)$} \label{comboexactfirst}

\item $\sin\left(\arcsin\left(-\dfrac{\sqrt{2}}{2}\right)\right)$

\item $\sin\left(\arcsin\left(\dfrac{3}{5}\right)\right)$ \vphantom{$\left( -\dfrac{\sqrt{2}}{2} \right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sin\left(\arcsin\left(-0.42\right)\right)$ \vphantom{$\left( \dfrac{\sqrt{2}}{2} \right)$}

\item $\sin\left(\arcsin\left(\dfrac{5}{4}\right)\right)$ \vphantom{$\left( \dfrac{\sqrt{2}}{2} \right)$}

\item $\cos\left(\arccos\left(\dfrac{\sqrt{2}}{2}\right)\right)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos\left(\arccos\left(-\dfrac{1}{2}\right)\right)$

\item $\cos\left(\arccos\left(\dfrac{5}{13}\right)\right)$

\item $\cos\left(\arccos\left(-0.998\right)\right)$ \vphantom{$\left( -\dfrac{1}{2} \right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos\left(\arccos\left(\pi \right)\right)$

\item $\tan\left(\arctan\left(-1\right)\right)$

\item $\tan\left(\arctan\left(\sqrt{3}\right)\right)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\tan\left(\arctan\left(\dfrac{5}{12}\right)\right)$

\item $\tan\left(\arctan\left(0.965\right)\right)$ \vphantom{$\left( \dfrac{1}{2} \right)$}

\item $\tan\left(\arctan\left( 3\pi \right)\right)$ \vphantom{$\left( \dfrac{1}{2} \right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cot\left(\text{arccot}\left(1\right)\right)$ \vphantom{$\left( \dfrac{1}{2} \right)$}

\item $\cot\left(\text{arccot}\left(-\sqrt{3}\right)\right)$ \vphantom{$\left( \dfrac{1}{2} \right)$}

\item $\cot\left(\text{arccot}\left(-\dfrac{7}{24}\right)\right)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cot\left(\text{arccot}\left(-0.001\right)\right)$ \vphantom{$\left( \dfrac{1}{2} \right)$}

\item $\cot\left(\text{arccot}\left( \dfrac{17\pi}{4} \right)\right)$

\item $\sec\left(\text{arcsec}\left(2\right)\right)$ \vphantom{$\left( \dfrac{1}{2} \right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sec\left(\text{arcsec}\left(-1\right)\right)$ \vphantom{$\left( \dfrac{1}{2} \right)$}

\item $\sec\left(\text{arcsec}\left(\dfrac{1}{2}\right)\right)$

\item $\sec\left(\text{arcsec}\left(0.75\right)\right)$ \vphantom{$\left( \dfrac{1}{2} \right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sec\left(\text{arcsec}\left( 117\pi \right)\right)$ \vphantom{$\left( \dfrac{\sqrt{3}}{3} \right)$}

\item $\csc\left(\text{arccsc}\left(\sqrt{2}\right)\right)$ \vphantom{$\left( \dfrac{\sqrt{3}}{3} \right)$}

\item $\csc\left(\text{arccsc}\left(-\dfrac{2\sqrt{3}}{3}\right)\right)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\csc\left(\text{arccsc}\left(\dfrac{\sqrt{2}}{2}\right)\right)$

\item $\csc\left(\text{arccsc}\left(1.0001\right)\right)$ \vphantom{$\left( \dfrac{\sqrt{3}}{3} \right)$}

\item $\csc\left(\text{arccsc}\left( \dfrac{\pi}{4} \right)\right)$ \vphantom{$\left( \dfrac{\sqrt{3}}{3} \right)$} \label{comboexactlast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

In Exercises \ref{morecomboexactfirst} - \ref{morecomboexactlast}, find the exact value or state that it is undefined.

\enlargethispage{.25in}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\arcsin\left(\sin\left(\dfrac{\pi}{6}\right) \right)$ \vphantom{$\left(\dfrac{3\pi}{4}\right)$} \label{morecomboexactfirst}

\item $\arcsin\left(\sin\left(-\dfrac{\pi}{3}\right) \right)$ \vphantom{$\left(\dfrac{3\pi}{4}\right)$}

\item $\arcsin\left(\sin\left(\dfrac{3\pi}{4}\right) \right)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\arcsin\left(\sin\left(\dfrac{11\pi}{6}\right) \right)$

\item $\arcsin\left(\sin\left(\dfrac{4\pi}{3}\right) \right)$

\item $\arccos\left(\cos\left(\dfrac{\pi}{4}\right) \right)$ \vphantom{$\left(\dfrac{3\pi}{4}\right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\arccos\left(\cos\left(\dfrac{2\pi}{3}\right) \right)$

\item $\arccos\left(\cos\left(\dfrac{3\pi}{2}\right) \right)$

\item $\arccos\left(\cos\left(-\dfrac{\pi}{6}\right) \right)$ \vphantom{$\left(\dfrac{3\pi}{4}\right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\arccos\left(\cos\left(\dfrac{5\pi}{4}\right) \right)$

\item $\arctan\left(\tan\left(\dfrac{\pi}{3}\right) \right)$ \vphantom{$\left(\dfrac{3\pi}{4}\right)$}

\item $\arctan\left(\tan\left(-\dfrac{\pi}{4}\right) \right)$ \vphantom{$\left(\dfrac{3\pi}{4}\right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\arctan\left(\tan\left(\pi\right) \right)$ \vphantom{$\left(\dfrac{3\pi}{4}\right)$}

\item $\arctan\left(\tan\left(\dfrac{\pi}{2}\right) \right)$ \vphantom{$\left(\dfrac{3\pi}{4}\right)$}

\item $\arctan\left(\tan\left(\dfrac{2\pi}{3}\right) \right)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\text{arccot}\left(\cot\left(\dfrac{\pi}{3}\right) \right)$

\item $\text{arccot}\left(\cot\left(-\dfrac{\pi}{4}\right) \right)$

\item $\text{arccot}\left(\cot\left(\pi\right) \right)$ \vphantom{$\left(\dfrac{\pi}{4}\right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\text{arccot}\left(\cot\left(\dfrac{\pi}{2}\right) \right)$ \vphantom{$\left(\dfrac{3\pi}{4}\right)$}

\item $\text{arccot}\left(\cot\left(\dfrac{2\pi}{3}\right) \right)$ \label{morecomboexactlast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

In Exercises \ref{extracombofirst} - \ref{extracombolast}, assume that the range of arcsecant is $\left[0, \frac{\pi}{2} \right) \cup \left[\pi, \frac{3\pi}{2} \right)$ and that the range of arccosecant is $\left(0, \frac{\pi}{2} \right] \cup \left( \pi, \frac{3\pi}{2} \right]$ when finding the exact value.

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\text{arcsec}\left(\sec\left(\dfrac{\pi}{4}\right) \right)$ \vphantom{$\left(\dfrac{4\pi}{3}\right)$} \label{extracombofirst}

\item $\text{arcsec}\left(\sec\left(\dfrac{4\pi}{3}\right) \right)$

\item $\text{arcsec}\left(\sec\left( \dfrac{5\pi}{6} \right) \right)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\text{arcsec}\left(\sec\left(-\dfrac{\pi}{2} \right) \right)$ \vphantom{$\left(\dfrac{4\pi}{3}\right)$}

\item $\text{arcsec}\left(\sec\left(\dfrac{5\pi}{3}\right) \right)$

\item $\text{arccsc}\left(\csc\left(\dfrac{\pi}{6}\right) \right)$ \vphantom{$\left(\dfrac{4\pi}{3}\right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\text{arccsc}\left(\csc\left(\dfrac{5\pi}{4}\right) \right)$

\item $\text{arccsc}\left(\csc\left( \dfrac{2\pi}{3} \right) \right)$

\item $\text{arccsc}\left(\csc\left(-\dfrac{\pi}{2} \right) \right)$ \vphantom{$\left(\dfrac{4\pi}{3}\right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\text{arccsc}\left(\csc\left(\dfrac{11\pi}{6}\right) \right)$

\item $\text{arcsec}\left(\sec\left(\dfrac{11\pi}{12}\right) \right)$

\item $\text{arccsc}\left(\csc\left(\dfrac{9\pi}{8}\right) \right)$ \label{extracombolast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

In Exercises \ref{moreextracombofirst} - \ref{moreextracombolast}, assume that the range of arcsecant is $\left[0, \frac{\pi}{2} \right) \cup \left( \frac{\pi}{2}, \pi \right]$ and that the range of arccosecant is $\left[ -\frac{\pi}{2}, 0 \right) \cup \left(0, \frac{\pi}{2} \right]$ when finding the exact value.

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\text{arcsec}\left(\sec\left(\dfrac{\pi}{4}\right) \right)$ \vphantom{$\left(\dfrac{4\pi}{3}\right)$} \label{moreextracombofirst}

\item $\text{arcsec}\left(\sec\left(\dfrac{4\pi}{3}\right) \right)$

\item $\text{arcsec}\left(\sec\left( \dfrac{5\pi}{6} \right) \right)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\text{arcsec}\left(\sec\left(-\dfrac{\pi}{2} \right) \right)$ \vphantom{$\left(\dfrac{4\pi}{3}\right)$}

\item $\text{arcsec}\left(\sec\left(\dfrac{5\pi}{3}\right) \right)$

\item $\text{arccsc}\left(\csc\left(\dfrac{\pi}{6}\right) \right)$ \vphantom{$\left(\dfrac{4\pi}{3}\right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\text{arccsc}\left(\csc\left(\dfrac{5\pi}{4}\right) \right)$

\item $\text{arccsc}\left(\csc\left( \dfrac{2\pi}{3} \right) \right)$

\item $\text{arccsc}\left(\csc\left(-\dfrac{\pi}{2} \right) \right)$ \vphantom{$\left(\dfrac{4\pi}{3}\right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\text{arccsc}\left(\csc\left(\dfrac{11\pi}{6}\right) \right)$

\item $\text{arcsec}\left(\sec\left(\dfrac{11\pi}{12}\right) \right)$

\item $\text{arccsc}\left(\csc\left(\dfrac{9\pi}{8}\right) \right)$ \label{moreextracombolast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\pagebreak

In Exercises \ref{stillmoreexactfirst} - \ref{stillmoreexactlast}, find the exact value or state that it is undefined.

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sin\left(\arccos\left(-\dfrac{1}{2}\right)\right)$ \label{stillmoreexactfirst}

\item $\sin\left(\arccos\left(\dfrac{3}{5}\right)\right)$

\item $\sin\left(\arctan\left(-2\right)\right)$ \vphantom{$\left(\dfrac{4}{3}\right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sin\left(\text{arccot}\left(\sqrt{5}\right)\right)$ \vphantom{$\left(\dfrac{4}{3}\right)$}

\item $\sin\left(\text{arccsc}\left(-3\right)\right)$ \vphantom{$\left(\dfrac{4}{3}\right)$}

\item $\cos\left(\arcsin\left(-\dfrac{5}{13}\right)\right)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos\left(\arctan\left(\sqrt{7} \right)\right)$

\item $\cos\left(\text{arccot}\left( 3 \right)\right)$

\item $\cos\left(\text{arcsec}\left( 5 \right)\right)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\tan\left(\arcsin\left(-\dfrac{2\sqrt{5}}{5}\right)\right)$

\item $\tan\left(\arccos\left(-\dfrac{1}{2}\right)\right)$ \vphantom{$\left(\dfrac{2\sqrt{2}}{3}\right)$}

\item $\tan\left(\text{arcsec}\left(\dfrac{5}{3}\right)\right)$ \vphantom{$\left(\dfrac{2\sqrt{2}}{3}\right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\tan\left(\text{arccot}\left( 12 \right)\right)$ \vphantom{$\left(\dfrac{2\sqrt{2}}{3}\right)$}

\item $\cot\left(\arcsin\left(\dfrac{12}{13}\right)\right)$ \vphantom{$\left(\dfrac{2\sqrt{2}}{3}\right)$}

\item $\cot\left(\arccos\left(\dfrac{\sqrt{3}}{2}\right)\right)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cot\left(\text{arccsc}\left(\sqrt{5}\right)\right)$ \vphantom{$\left(\dfrac{2\sqrt{2}}{3}\right)$}

\item $\cot\left(\arctan \left( 0.25 \right)\right)$ \vphantom{$\left(\dfrac{2\sqrt{2}}{3}\right)$}

\item $\sec\left(\arccos\left(\dfrac{\sqrt{3}}{2}\right)\right)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sec\left(\arcsin\left(-\dfrac{12}{13}\right)\right)$ \vphantom{$\left(\dfrac{2\sqrt{2}}{3}\right)$}

\item $\sec\left(\arctan\left(10\right)\right)$ \vphantom{$\left(\dfrac{2\sqrt{2}}{3}\right)$}

\item $\sec\left(\text{arccot}\left(-\dfrac{\sqrt{10}}{10}\right)\right)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\csc\left(\text{arccot}\left(9 \right)\right)$ \vphantom{$\left(\dfrac{2}{3}\right)$}

\item $\csc\left(\arcsin\left(\dfrac{3}{5}\right)\right)$

\item $\csc\left(\arctan\left(-\dfrac{2}{3}\right)\right)$ \label{stillmoreexactlast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

In Exercises \ref{exactvalueidenfirst} - \ref{exactvalueidenlast}, find the exact value or state that it is undefined.

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sin\left(\arcsin\left( \dfrac{5}{13} \right) + \dfrac{\pi}{4}\right)$ \label{exactvalueidenfirst}

\item $\cos\left( \text{arcsec}(3) + \arctan(2) \right)$ \vphantom{$\left(\dfrac{2}{3}\right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\tan\left( \arctan(3) + \arccos\left(-\dfrac{3}{5}\right) \right)$

\item $\sin\left(2\arcsin\left(-\dfrac{4}{5}\right)\right)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sin\left(2\text{arccsc}\left(\dfrac{13}{5}\right)\right)$

\item $\sin\left(2\arctan\left(2\right)\right)$ \vphantom{$\left(\dfrac{2}{3}\right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos\left(2 \arcsin\left(\dfrac{3}{5}\right)\right)$

\item $\cos\left(2 \text{arcsec}\left(\dfrac{25}{7}\right)\right)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos\left(2 \text{arccot}\left(-\sqrt{5}\right)\right)$ \vphantom{$\left(\dfrac{2}{3}\right)$}

\item $\sin\left( \dfrac{\arctan(2)}{2} \right)$ \label{exactvalueidenlast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\pagebreak

In Exercises \ref{rewritefirst} - \ref{rewritelast}, rewrite the quantity as algebraic expressions of $x$ and state the domain on which the equivalence is valid.

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sin \left( \arccos \left( x \right) \right)$ \label{rewritefirst}

\item $\cos \left( \arctan \left( x \right) \right)$

\item $\tan \left( \arcsin \left( x \right) \right)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sec \left( \arctan \left( x \right) \right)$

\item $\csc \left( \arccos \left( x \right) \right)$

\item $\sin \left( 2\arctan \left( x \right) \right)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sin \left( 2\arccos \left( x \right) \right)$

\item $\cos \left( 2\arctan \left( x \right) \right)$

\item $\sin(\arccos(2x))$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sin\left(\arccos\left(\dfrac{x}{5}\right)\right)$

\item $\cos\left(\arcsin\left(\dfrac{x}{2}\right)\right)$

\item $\cos\left(\arctan\left(3x\right)\right)$ \vphantom{$\left(\dfrac{x}{5}\right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sin(2\arcsin(7x))$ \vphantom{$\left(\dfrac{x\sqrt{3}}{5}\right)$}

\item $\sin\left(2 \arcsin\left( \dfrac{x\sqrt{3}}{3} \right) \right)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos(2 \arcsin(4x))$

\item $\sec(\arctan(2x))\tan(\arctan(2x))$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sin \left( \arcsin(x) + \arccos(x) \right)$

\item $\cos \left( \arcsin(x) + \arctan(x) \right)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\tan \left( 2\arcsin(x) \right)$ \vphantom{$\left(\dfrac{1}{2}\right)$}

\item $\sin \left( \dfrac{1}{2}\arctan(x) \right)$ \label{rewritelast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item If $\sin(\theta) = \dfrac{x}{2}$ for $-\dfrac{\pi}{2} < \theta < \dfrac{\pi}{2}$, find an expression for $\theta + \sin(2\theta)$ in terms of $x$.

\item If $\tan(\theta) = \dfrac{x}{7}$ for $-\dfrac{\pi}{2} < \theta < \dfrac{\pi}{2}$, find an expression for $\dfrac{1}{2}\theta - \dfrac{1}{2}\sin(2\theta)$ in terms of $x$.

\item If $\sec(\theta) = \dfrac{x}{4}$ for $0 < \theta < \dfrac{\pi}{2}$, find an expression for $4\tan(\theta) - 4\theta$ in terms of $x$.

\setcounter{HW}{\value{enumi}}

\end{enumerate}

In Exercises \ref{equarctrigfirst} - \ref{equarctriglast}, solve the equation using the techniques discussed in Example \ref{basicinverseeqns} then approximate the solutions which lie in the interval $[0, 2\pi)$ to four decimal places.

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sin(x) = \dfrac{7}{11}$ \label{equarctrigfirst}

\item $\cos(x) = -\dfrac{2}{9}$

\item $\sin(x) = -0.569$ \vphantom{$\dfrac{1}{2}$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos(x) = 0.117$ \vphantom{$\dfrac{1}{2}$}

\item $\sin(x) = 0.008$ \vphantom{$\dfrac{1}{2}$}

\item $\cos(x) = \dfrac{359}{360}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\tan(x) = 117$ \vphantom{$\dfrac{1}{2}$}

\item $\cot(x) = -12$ \vphantom{$\dfrac{1}{2}$}

\item $\sec(x) = \dfrac{3}{2}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\csc(x) = -\dfrac{90}{17}$

\item $\tan(x) = -\sqrt{10}$ \vphantom{$\dfrac{1}{2}$}

\item $\sin(x) = \dfrac{3}{8}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos(x) = -\dfrac{7}{16}$

\item $\tan(x) = 0.03$ \vphantom{$\dfrac{1}{2}$}

\item $\sin(x) = 0.3502$ \vphantom{$\dfrac{1}{2}$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sin(x) = -0.721$

\item $\cos(x) = 0.9824$

\item $\cos(x) = -0.5637$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cot(x) = \dfrac{1}{117}$

\item $\tan(x) = -0.6109$ \vphantom{$\dfrac{1}{2}$} \label{equarctriglast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

In Exercises \ref{trianglesidesfirst} - \ref{trianglesideslast}, find the two acute angles in the right triangle whose sides have the given lengths. Express your answers using degree measure rounded to two decimal places.

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item 3, 4 and 5 \label{trianglesidesfirst}

\item 5, 12 and 13

\item 336, 527 and 625 \label{trianglesideslast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item A guy wire 1000 feet long is attached to the top of a tower. When pulled taut it touches level ground 360 feet from the base of the tower. What angle does the wire make with the ground? Express your answer using degree measure rounded to one decimal place.

\item At Cliffs of Insanity Point, The Great Sasquatch Canyon is 7117 feet deep. From that point, a fire is seen at a location known to be 10 miles away from the base of the sheer canyon wall. What angle of depression is made by the line of sight from the canyon edge to the fire? Express your answer using degree measure rounded to one decimal place.

\item Shelving is being built at the Utility Muffin Research Library which is to be 14 inches deep. An 18-inch rod will be attached to the wall and the underside of the shelf at its edge away from the wall, forming a right triangle under the shelf to support it. What angle, to the nearest degree, will the rod make with the wall?

\item A parasailor is being pulled by a boat on Lake Ippizuti. The cable is 300 feet long and the parasailor is 100 feet above the surface of the water. What is the angle of elevation from the boat to the parasailor? Express your answer using degree measure rounded to one decimal place.

\item A tag-and-release program to study the Sasquatch population of the eponymous Sasquatch National Park is begun. From a 200 foot tall tower, a ranger spots a Sasquatch lumbering through the wilderness directly towards the tower. Let $\theta$ denote the angle of depression from the top of the tower to a point on the ground. If the range of the rifle with a tranquilizer dart is 300 feet, find the smallest value of $\theta$ for which the corresponding point on the ground is in range of the rifle. Round your answer to the nearest hundreth of a degree.

\setcounter{HW}{\value{enumi}}

\end{enumerate}

In Exercises \ref{rewritesinusoidfirst} - \ref{rewritesinusoidlast}, rewrite the given function as a sinusoid of the form $S(x) = A\sin(\omega x + \phi)$ using Exercises \ref{sinusoidexercise1} and \ref{sinusoidexercise2} in Section \ref{TrigGraphs} for reference. Approximate the value of $\phi$ (which is in radians, of course) to four decimal places.

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = 5\sin(3x) + 12\cos(3x)$ \label{rewritesinusoidfirst}

\item $f(x) = 3\cos(2x) + 4\sin(2x)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = \cos(x) - 3\sin(x)$

\item $f(x) = 7\sin(10x) - 24\cos(10x)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = -\cos(x) - 2\sqrt{2} \sin(x)$

\item $f(x) = 2\sin(x) - \cos(x)$ \label{rewritesinusoidlast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

In Exercises \ref{domainexerfirst} - \ref{domainexerlast}, find the domain of the given function. Write your answers in interval notation.

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = \arcsin(5x)$ \vphantom{$\left(\dfrac{3x-1}{2} \right)$} \label{domainexerfirst}

\item $f(x) = \arccos\left(\dfrac{3x-1}{2} \right)$

\item $f(x) = \arcsin\left(2x^2\right)$ \vphantom{$\left(\dfrac{3x-1}{2} \right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = \arccos\left(\dfrac{1}{x^2-4}\right)$

\item $f(x) = \arctan(4x)$ \vphantom{$\left(\dfrac{3x-1}{2} \right)$}

\item $f(x) = \text{arccot}\left(\dfrac{2x}{x^2-9}\right)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) =\arctan(\ln(2x-1))$

\item $f(x) = \text{arccot}(\sqrt{2x-1})$

\item $f(x) = \text{arcsec}(12x)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = \text{arccsc}(x+5)$ \vphantom{$\left(\dfrac{3x-1}{2} \right)$}

\item $f(x) = \text{arcsec}\left(\dfrac{x^3}{8}\right)$

\item $f(x) = \text{arccsc}\left(e^{2x}\right)$ \vphantom{$\left(\dfrac{3x-1}{2} \right)$} \label{domainexerlast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item Show that $\mbox{arcsec}(x) = \arccos \left( \dfrac{1}{x} \right)$ for $|x| \geq 1$ as long as we use $\left[0, \dfrac{\pi}{2} \right) \cup \left( \dfrac{\pi}{2}, \pi \right]$ as the range of $f(x) = \mbox{arcsec}(x)$.

\item Show that $\mbox{arccsc}(x) = \arcsin \left( \dfrac{1}{x} \right)$ for $|x| \geq 1$ as long as we use $\left[ -\dfrac{\pi}{2}, 0 \right) \cup \left(0, \dfrac{\pi}{2} \right]$ as the range of $f(x) = \mbox{arccsc}(x)$.

\item Show that $\arcsin(x) + \arccos(x) = \dfrac{\pi}{2}$ for $-1 \leq x \leq 1$.

\item Discuss with your classmates why $\arcsin\left(\dfrac{1}{2}\right) \neq 30^{\circ}$.

\item Use the following picture and the series of exercises on the next page to show that \[\arctan(1) + \arctan(2) + \arctan(3) = \pi\]

\begin{center}

\begin{mfpic}[50]{-1}{2.25}{0}{3.25}

\axes

\point[3pt]{(0,0), (1,0), (2,0), (2,3), (0,1)}

\tlabel(2.35,0){\scriptsize $x$}

\tlabel(0.15,3.25){\scriptsize $y$}

\tlabel(-0.8,0.9){$A(0,1)$}

\tlabel(-0.25,-0.25){$O(0,0)$}

\tlabel(0.75,-0.25){$B(1,0)$}

\tlabel(1.75,-0.25){$C(2,0)$}

\tlabel(2.05,3){$D(2,3)$}

\polyline{(0,1), (1,0)}

\polyline{(1,0), (2,3)}

\polyline{(0,1), (2,3)}

\polyline{(2,0), (2,3)}

\tlabel(0.65,0.05){\small $\alpha$}

\tlabel(0.88,0.15){\small $\beta$}

\tlabel(1.15,0.08){\small $\gamma$}

\end{mfpic}

\end{center}

\begin{enumerate}

\item Clearly $\triangle AOB$ and $\triangle BCD$ are right triangles because the line through $O$ and $A$ and the line through $C$ and $D$ are perpendicular to the $x$-axis. Use the distance formula to show that $\triangle BAD$ is also a right triangle (with $\angle BAD$ being the right angle) by showing that the sides of the triangle satisfy the Pythagorean Theorem.

\item Use $\triangle AOB$ to show that $\alpha = \arctan(1)$

\item Use $\triangle BAD$ to show that $\beta = \arctan(2)$

\item Use $\triangle BCD$ to show that $\gamma = \arctan(3)$

\item Use the fact that $O$, $B$ and $C$ all lie on the $x$-axis to conclude that $\alpha + \beta + \gamma = \pi$. Thus $\arctan(1) + \arctan(2) + \arctan(3) = \pi$.

\end{enumerate}

\end{enumerate}

\newpage

\subsection{Answers}

\begin{multicols}{3}

\begin{enumerate}

\item $\arcsin \left( -1 \right) = -\dfrac{\pi}{2}$ \vphantom{$\left( -\dfrac{\sqrt{3}}{2} \right)$}

\item $\arcsin \left( -\dfrac{\sqrt{3}}{2} \right) = -\dfrac{\pi}{3}$

\item $\arcsin \left( -\dfrac{\sqrt{2}}{2} \right) = -\dfrac{\pi}{4}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\arcsin \left( -\dfrac{1}{2} \right) = -\dfrac{\pi}{6}$

\item $\arcsin \left( 0 \right) = 0$ \vphantom{$\left( -\dfrac{1}{2} \right)$}

\item $\arcsin \left( \dfrac{1}{2} \right) = \dfrac{\pi}{6}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\arcsin \left( \dfrac{\sqrt{2}}{2} \right) = \dfrac{\pi}{4}$

\item $\arcsin \left( \dfrac{\sqrt{3}}{2} \right) = \dfrac{\pi}{3}$

\item $\arcsin \left( 1 \right) = \dfrac{\pi}{2}$ \vphantom{$\left( -\dfrac{\sqrt{3}}{2} \right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\arccos \left( -1 \right) = \pi$ \vphantom{$\left( -\dfrac{\sqrt{3}}{2} \right)$}

\item $\arccos \left( -\dfrac{\sqrt{3}}{2} \right) = \dfrac{5\pi}{6}$

\item $\arccos \left( -\dfrac{\sqrt{2}}{2} \right) = \dfrac{3\pi}{4}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\arccos \left( -\dfrac{1}{2} \right) = \dfrac{2\pi}{3}$

\item $\arccos \left( 0 \right) = \dfrac{\pi}{2}$ \vphantom{$\left( -\dfrac{1}{2} \right)$}

\item $\arccos \left( \dfrac{1}{2} \right) = \dfrac{\pi}{3}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\arccos \left( \dfrac{\sqrt{2}}{2} \right) = \dfrac{\pi}{4}$

\item $\arccos \left( \dfrac{\sqrt{3}}{2} \right) = \dfrac{\pi}{6}$

\item $\arccos \left( 1 \right) = 0$ \vphantom{$\left( -\dfrac{\sqrt{3}}{2} \right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\arctan \left( -\sqrt{3} \right) = -\dfrac{\pi}{3}$ \vphantom{$\left( -\dfrac{\sqrt{3}}{2} \right)$}

\item $\arctan \left( -1 \right) = -\dfrac{\pi}{4}$ \vphantom{$\left( -\dfrac{\sqrt{3}}{2} \right)$}

\item $\arctan \left( -\dfrac{\sqrt{3}}{3} \right) = -\dfrac{\pi}{6}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\arctan \left( 0 \right) = 0$ \vphantom{$\left( -\dfrac{\sqrt{3}}{2} \right)$}

\item $\arctan \left( \dfrac{\sqrt{3}}{3} \right) = \dfrac{\pi}{6}$

\item $\arctan \left( 1 \right) = \dfrac{\pi}{4}$ \vphantom{$\left( -\dfrac{\sqrt{3}}{2} \right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\arctan \left( \sqrt{3} \right) = \dfrac{\pi}{3}$ \vphantom{$\dfrac{3\pi}{2}$}

\item $\mbox{arccot} \left( -\sqrt{3} \right) = \dfrac{5\pi}{6}$

\item $\mbox{arccot} \left( -1 \right) = \dfrac{3\pi}{4}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\mbox{arccot} \left( -\dfrac{\sqrt{3}}{3} \right) = \dfrac{2\pi}{3}$

\item $\mbox{arccot} \left( 0 \right) = \dfrac{\pi}{2}$ \vphantom{$\left( -\dfrac{\sqrt{3}}{2} \right)$}

\item $\mbox{arccot} \left( \dfrac{\sqrt{3}}{3} \right) = \dfrac{\pi}{3}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\mbox{arccot} \left( 1 \right) = \dfrac{\pi}{4}$

\item $\mbox{arccot} \left( \sqrt{3} \right) = \dfrac{\pi}{6}$

\item $\mbox{arcsec} \left( 2 \right) = \dfrac{\pi}{3}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\mbox{arccsc} \left( 2 \right) = \dfrac{\pi}{6}$

\item $\mbox{arcsec} \left( \sqrt{2} \right) = \dfrac{\pi}{4}$

\item $\mbox{arccsc} \left( \sqrt{2} \right) = \dfrac{\pi}{4}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\mbox{arcsec} \left( \dfrac{2\sqrt{3}}{3} \right) = \dfrac{\pi}{6}$

\item $\mbox{arccsc} \left( \dfrac{2\sqrt{3}}{3} \right) = \dfrac{\pi}{3}$

\item $\mbox{arcsec} \left( 1 \right) = 0$ \vphantom{$\left( \dfrac{2\sqrt{3}}{2} \right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\mbox{arccsc} \left( 1 \right) = \dfrac{\pi}{2}$ \vphantom{$\dfrac{3\pi}{2}$}

\item $\mbox{arcsec} \left( -2 \right) = \dfrac{4\pi}{3}$

\item $\mbox{arcsec} \left( -\sqrt{2} \right) = \dfrac{5\pi}{4}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\mbox{arcsec} \left( -\dfrac{2\sqrt{3}}{3} \right) = \dfrac{7\pi}{6}$

\item $\mbox{arcsec} \left( -1 \right) = \pi$ \vphantom{$\left( -\dfrac{\sqrt{3}}{2} \right)$}

\item $\mbox{arccsc} \left( -2 \right) = \dfrac{7\pi}{6}$ \vphantom{$\left( -\dfrac{\sqrt{3}}{2} \right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\mbox{arccsc} \left( -\sqrt{2} \right) = \dfrac{5\pi}{4}$ \vphantom{$\left( -\dfrac{2\sqrt{3}}{2} \right)$}

\item $\mbox{arccsc} \left( -\dfrac{2\sqrt{3}}{3} \right) = \dfrac{4\pi}{3}$

\item $\mbox{arccsc} \left( -1 \right) = \dfrac{3\pi}{2}$ \vphantom{$\left( -\dfrac{2\sqrt{3}}{2} \right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\mbox{arcsec} \left( -2 \right) = \dfrac{2\pi}{3}$ \vphantom{$\left( -\dfrac{2\sqrt{3}}{2} \right)$}

\item $\mbox{arcsec} \left( -\sqrt{2} \right) = \dfrac{3\pi}{4}$ \vphantom{$\left( -\dfrac{2\sqrt{3}}{2} \right)$}

\item $\mbox{arcsec} \left( -\dfrac{2\sqrt{3}}{3} \right) = \dfrac{5\pi}{6}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\mbox{arcsec} \left( -1 \right) = \pi$ \vphantom{$-\dfrac{\pi}{2}$}

\item $\mbox{arccsc} \left( -2 \right) = -\dfrac{\pi}{6}$

\item $\mbox{arccsc} \left( -\sqrt{2} \right) = -\dfrac{\pi}{4}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\mbox{arccsc} \left( -\dfrac{2\sqrt{3}}{3} \right) = -\dfrac{\pi}{3}$

\item $\mbox{arccsc} \left( -1 \right) = -\dfrac{\pi}{2}$ \vphantom{$\left( -\dfrac{\sqrt{3}}{2} \right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sin\left(\arcsin\left(\dfrac{1}{2}\right)\right) = \dfrac{1}{2}$ \vphantom{$\left(-\dfrac{\sqrt{2}}{2}\right)$}

\item $\sin\left(\arcsin\left(-\dfrac{\sqrt{2}}{2}\right)\right) = -\dfrac{\sqrt{2}}{2}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sin\left(\arcsin\left(\dfrac{3}{5}\right)\right) = \dfrac{3}{5}$

\item $\sin\left(\arcsin\left(-0.42\right)\right) = -0.42$ \vphantom{$\left(-\dfrac{3}{2}\right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sin\left(\arcsin\left(\dfrac{5}{4}\right)\right)$ is undefined. \vphantom{$\left(-\dfrac{\sqrt{2}}{2}\right)$}

\item $\cos\left(\arccos\left(\dfrac{\sqrt{2}}{2}\right)\right) = \dfrac{\sqrt{2}}{2}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos\left(\arccos\left(-\dfrac{1}{2}\right)\right) = -\dfrac{1}{2}$

\item $\cos\left(\arccos\left(\dfrac{5}{13}\right)\right) = \dfrac{5}{13}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos\left(\arccos\left(-0.998\right)\right) = -0.998$

\item $\cos\left(\arccos\left(\pi \right)\right)$ is undefined.

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\tan\left(\arctan\left(-1\right)\right) = -1$

\item $\tan\left(\arctan\left(\sqrt{3}\right)\right) = \sqrt{3}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\tan\left(\arctan\left(\dfrac{5}{12}\right)\right) = \dfrac{5}{12}$

\item $\tan\left(\arctan\left(0.965\right)\right) = 0.965$ \vphantom{$\left(-\dfrac{2}{2}\right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\tan\left(\arctan\left( 3\pi \right)\right) = 3\pi$

\item $\cot\left(\text{arccot}\left(1\right)\right) = 1$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cot\left(\text{arccot}\left(-\sqrt{3}\right)\right) = -\sqrt{3}$ \vphantom{$\left(-\dfrac{1}{2}\right)$}

\item $\cot\left(\text{arccot}\left(-\dfrac{7}{24}\right)\right) = -\dfrac{7}{24}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cot\left(\text{arccot}\left(-0.001\right)\right) = -0.001$ \vphantom{$\left(-\dfrac{7\pi}{2}\right)$}

\item $\cot\left(\text{arccot}\left( \dfrac{17\pi}{4} \right)\right) = \dfrac{17\pi}{4}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sec\left(\text{arcsec}\left(2\right)\right) = 2$

\item $\sec\left(\text{arcsec}\left(-1\right)\right) = -1$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sec\left(\text{arcsec}\left(\dfrac{1}{2}\right)\right)$ is undefined.

\item $\sec\left(\text{arcsec}\left(0.75\right)\right)$ is undefined. \vphantom{$\left(-\dfrac{1}{2}\right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sec\left(\text{arcsec}\left( \dfrac{\pi}{2} \right)\right)= \dfrac{\pi}{2} $

\item $\csc\left(\text{arccsc}\left(\sqrt{2}\right)\right) = \sqrt{2}$ \vphantom{$\left(-\dfrac{\pi}{2}\right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\csc\left(\text{arccsc}\left(-\dfrac{2\sqrt{3}}{3}\right)\right) = -\dfrac{2\sqrt{3}}{3}$

\item $\csc\left(\text{arccsc}\left(\dfrac{\sqrt{2}}{2}\right)\right)$ is undefined.

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\csc\left(\text{arccsc}\left(1.0001\right)\right) = 1.0001$ \vphantom{$\left(-\dfrac{\pi}{2}\right)$}

\item $\csc\left(\text{arccsc}\left( \dfrac{\pi}{4} \right)\right)$ is undefined.

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\arcsin\left(\sin\left(\dfrac{\pi}{6}\right) \right) = \dfrac{\pi}{6}$

\item $\arcsin\left(\sin\left(-\dfrac{\pi}{3}\right) \right) = -\dfrac{\pi}{3}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\arcsin\left(\sin\left(\dfrac{3\pi}{4}\right) \right) = \dfrac{\pi}{4}$

\item $\arcsin\left(\sin\left(\dfrac{11\pi}{6}\right) \right) = -\dfrac{\pi}{6}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\arcsin\left(\sin\left(\dfrac{4\pi}{3}\right) \right) = -\dfrac{\pi}{3}$

\item $\arccos\left(\cos\left(\dfrac{\pi}{4}\right) \right) = \dfrac{\pi}{4}$ \vphantom{$\left(-\dfrac{3\pi}{2}\right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\arccos\left(\cos\left(\dfrac{2\pi}{3}\right) \right) = \dfrac{2\pi}{3}$

\item $\arccos\left(\cos\left(\dfrac{3\pi}{2}\right) \right) = \dfrac{\pi}{2}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\arccos\left(\cos\left(-\dfrac{\pi}{6}\right) \right) = \dfrac{\pi}{6}$ \vphantom{$\left(-\dfrac{6\pi}{2}\right)$}

\item $\arccos\left(\cos\left(\dfrac{5\pi}{4}\right) \right) = \dfrac{3\pi}{4}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\arctan\left(\tan\left(\dfrac{\pi}{3}\right) \right) = \dfrac{\pi}{3}$

\item $\arctan\left(\tan\left(-\dfrac{\pi}{4}\right) \right) = -\dfrac{\pi}{4}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\arctan\left(\tan\left(\pi\right) \right) = 0$ \vphantom{$\left(-\dfrac{\pi}{2}\right)$}

\item $\arctan\left(\tan\left(\dfrac{\pi}{2}\right) \right)$ is undefined

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\arctan\left(\tan\left(\dfrac{2\pi}{3}\right) \right) = -\dfrac{\pi}{3}$

\item $\text{arccot}\left(\cot\left(\dfrac{\pi}{3}\right) \right) = \dfrac{\pi}{3}$ \vphantom{$\left(-\dfrac{3\pi}{2}\right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\text{arccot}\left(\cot\left(-\dfrac{\pi}{4}\right) \right) = \dfrac{3\pi}{4}$

\item $\text{arccot}\left(\cot\left(\pi\right) \right)$ is undefined \vphantom{$\left(-\dfrac{\pi}{2}\right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\text{arccot}\left(\cot\left(\dfrac{3\pi}{2}\right) \right) = \dfrac{\pi}{2}$

\item $\text{arccot}\left(\cot\left(\dfrac{2\pi}{3}\right) \right) = \dfrac{2\pi}{3}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\text{arcsec}\left(\sec\left(\dfrac{\pi}{4}\right) \right) = \dfrac{\pi}{4}$ \vphantom{$\left(-\dfrac{4\pi}{2}\right)$}

\item $\text{arcsec}\left(\sec\left(\dfrac{4\pi}{3}\right) \right) = \dfrac{4\pi}{3}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\text{arcsec}\left(\sec\left( \dfrac{5\pi}{6} \right) \right) = \dfrac{7\pi}{6}$

\item $\text{arcsec}\left(\sec\left(-\dfrac{\pi}{2} \right) \right)$ is undefined. \vphantom{$\left(-\dfrac{4\pi}{2}\right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\text{arcsec}\left(\sec\left(\dfrac{5\pi}{3}\right) \right) = \dfrac{\pi}{3}$

\item $\text{arccsc}\left(\csc\left(\dfrac{\pi}{6}\right) \right) = \dfrac{\pi}{6}$ \vphantom{$\left(-\dfrac{4\pi}{2}\right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\text{arccsc}\left(\csc\left(\dfrac{5\pi}{4}\right) \right) = \dfrac{5\pi}{4}$

\item $\text{arccsc}\left(\csc\left( \dfrac{2\pi}{3} \right) \right) = \dfrac{\pi}{3}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\text{arccsc}\left(\csc\left(-\dfrac{\pi}{2} \right) \right) = \dfrac{3\pi}{2}$ \vphantom{$\left(-\dfrac{4\pi}{2}\right)$}

\item $\text{arccsc}\left(\csc\left(\dfrac{11\pi}{6}\right) \right) = \dfrac{7\pi}{6}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\text{arcsec}\left(\sec\left(\dfrac{11\pi}{12}\right) \right) = \dfrac{13\pi}{12}$

\item $\text{arccsc}\left(\csc\left(\dfrac{9\pi}{8}\right) \right) = \dfrac{9\pi}{8}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\text{arcsec}\left(\sec\left(\dfrac{\pi}{4}\right) \right) = \dfrac{\pi}{4}$ \vphantom{$\left(-\dfrac{4\pi}{2}\right)$}

\item $\text{arcsec}\left(\sec\left(\dfrac{4\pi}{3}\right) \right) = \dfrac{2\pi}{3}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\text{arcsec}\left(\sec\left( \dfrac{5\pi}{6} \right) \right) = \dfrac{5\pi}{6}$

\item $\text{arcsec}\left(\sec\left(-\dfrac{\pi}{2} \right) \right)$ is undefined. \vphantom{$\left(-\dfrac{5\pi}{2}\right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\text{arcsec}\left(\sec\left(\dfrac{5\pi}{3}\right) \right) = \dfrac{\pi}{3}$

\item $\text{arccsc}\left(\csc\left(\dfrac{\pi}{6}\right) \right) = \dfrac{\pi}{6}$ \vphantom{$\left(-\dfrac{5\pi}{2}\right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\text{arccsc}\left(\csc\left(\dfrac{5\pi}{4}\right) \right) = -\dfrac{\pi}{4}$

\item $\text{arccsc}\left(\csc\left( \dfrac{2\pi}{3} \right) \right) = \dfrac{\pi}{3}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\text{arccsc}\left(\csc\left(-\dfrac{\pi}{2} \right) \right) = -\dfrac{\pi}{2}$ \vphantom{$\left(-\dfrac{5\pi}{2}\right)$}

\item $\text{arccsc}\left(\csc\left(\dfrac{11\pi}{6}\right) \right) = -\dfrac{\pi}{6}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\text{arcsec}\left(\sec\left(\dfrac{11\pi}{12}\right) \right) = \dfrac{11\pi}{12}$

\item $\text{arccsc}\left(\csc\left(\dfrac{9\pi}{8}\right) \right) = -\dfrac{\pi}{8}$ \vphantom{$\left(-\dfrac{5\pi}{2}\right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sin\left(\arccos\left(-\dfrac{1}{2}\right)\right) = \dfrac{\sqrt{3}}{2}$

\item $\sin\left(\arccos\left(\dfrac{3}{5}\right)\right) = \dfrac{4}{5}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sin\left(\arctan\left(-2\right)\right) = -\dfrac{2\sqrt{5}}{5}$

\item $\sin\left(\text{arccot}\left(\sqrt{5}\right)\right) = \dfrac{\sqrt{6}}{6}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sin\left(\text{arccsc}\left(-3\right)\right) = -\dfrac{1}{3}$ \vphantom{$\left(-\dfrac{5}{2}\right)$}

\item $\cos\left(\arcsin\left(-\dfrac{5}{13}\right)\right) = \dfrac{12}{13}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos\left(\arctan\left(\sqrt{7} \right)\right) = \dfrac{\sqrt{2}}{4}$

\item $\cos\left(\text{arccot}\left( 3 \right)\right) = \dfrac{3\sqrt{10}}{10}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos\left(\text{arcsec}\left( 5 \right)\right) = \dfrac{1}{5}$ \vphantom{$\left(-\dfrac{5\sqrt{3}}{2}\right)$}

\item $\tan\left(\arcsin\left(-\dfrac{2\sqrt{5}}{5}\right)\right)=-2$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\tan\left(\arccos\left(-\dfrac{1}{2}\right)\right) = -\sqrt{3}$

\item $\tan\left(\text{arcsec}\left(\dfrac{5}{3}\right)\right) = \dfrac{4}{3}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\tan\left(\text{arccot}\left( 12 \right)\right) = \dfrac{1}{12}$ \vphantom{$\left(-\dfrac{5}{2}\right)$}

\item $\cot\left(\arcsin\left(\dfrac{12}{13}\right)\right) = \dfrac{5}{12}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cot\left(\arccos\left(\dfrac{\sqrt{3}}{2}\right)\right) = \sqrt{3}$

\item $\cot\left(\text{arccsc}\left(\sqrt{5}\right)\right) = 2$ \vphantom{$\left(-\dfrac{\sqrt{3}}{2}\right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cot\left(\arctan \left( 0.25 \right)\right) = 4$ \vphantom{$\left(-\dfrac{\sqrt{3}}{2}\right)$}

\item $\sec\left(\arccos\left(\dfrac{\sqrt{3}}{2}\right)\right) = \dfrac{2\sqrt{3}}{3}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sec\left(\arcsin\left(-\dfrac{12}{13}\right)\right) = \dfrac{13}{5}$

\item $\sec\left(\arctan\left(10\right)\right) = \sqrt{101}$ \vphantom{$\left(-\dfrac{5}{2}\right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sec\left(\text{arccot}\left(-\dfrac{\sqrt{10}}{10}\right)\right) = -\sqrt{11}$

\item $\csc\left(\text{arccot}\left(9 \right)\right) = \sqrt{82}$ \vphantom{$\left(-\dfrac{5\sqrt{3}}{2}\right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\csc\left(\arcsin\left(\dfrac{3}{5}\right)\right) = \dfrac{5}{3}$

\item $\csc\left(\arctan\left(-\dfrac{2}{3}\right)\right) = -\dfrac{\sqrt{13}}{2}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sin\left(\arcsin\left( \dfrac{5}{13} \right) + \dfrac{\pi}{4}\right) = \dfrac{17\sqrt{2}}{26}$

\item $\cos\left( \text{arcsec}(3) + \arctan(2) \right) = \dfrac{\sqrt{5} - 4\sqrt{10}}{15}$ %\vphantom{$\left(-\dfrac{5}{2}\right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\tan\left( \arctan(3) + \arccos\left(-\dfrac{3}{5}\right) \right) = \dfrac{1}{3}$

\item $\sin\left(2\arcsin\left(-\dfrac{4}{5}\right)\right)= -\dfrac{24}{25}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sin\left(2\text{arccsc}\left(\dfrac{13}{5}\right)\right) = \dfrac{120}{169}$

\item $\sin\left(2\arctan\left(2\right)\right) = \dfrac{4}{5}$ \vphantom{$\left(-\dfrac{5}{2}\right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos\left(2 \arcsin\left(\dfrac{3}{5}\right)\right) = \dfrac{7}{25}$

\item $\cos\left(2 \text{arcsec}\left(\dfrac{25}{7}\right)\right) = -\dfrac{527}{625}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos\left(2 \text{arccot}\left(-\sqrt{5}\right)\right) = \dfrac{2}{3}$ \vphantom{$\sqrt{\dfrac{5-\sqrt{5}}{10}}$}

\item $\sin\left( \dfrac{\arctan(2)}{2} \right) = \sqrt{\dfrac{5-\sqrt{5}}{10}}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sin \left( \arccos \left( x \right) \right) = \sqrt{1 - x^{2}}$ for $-1 \leq x \leq 1$

\item $\cos \left( \arctan \left( x \right) \right) = \dfrac{1}{\sqrt{1 + x^{2}}}$ for all $x$

\item $\tan \left( \arcsin \left( x \right) \right) = \dfrac{x}{\sqrt{1 - x^{2}}}$ for $-1 < x < 1$

\item $\sec \left( \arctan \left( x \right) \right) = \sqrt{1 + x^{2}}$ for all $x$

\item $\csc \left( \arccos \left( x \right) \right) = \dfrac{1}{\sqrt{1 - x^{2}}}$ for $-1 < x < 1$

\item $\sin \left( 2\arctan \left( x \right) \right) = \dfrac{2x}{x^{2} + 1}$ for all $x$

\item $\sin \left( 2\arccos \left( x \right) \right) = 2x\sqrt{1-x^2}$ for $-1 \leq x \leq 1$

\item $\cos \left( 2\arctan \left( x \right) \right) = \dfrac{1 - x^{2}}{1 + x^{2}}$ for all $x$

\item $\sin(\arccos(2x)) = \sqrt{1-4x^2}$ for $-\frac{1}{2} \leq x \leq \frac{1}{2}$

\item $\sin\left(\arccos\left(\dfrac{x}{5}\right)\right) = \dfrac{\sqrt{25-x^2}}{5}$ for $-5 \leq x \leq 5$

\item $\cos\left(\arcsin\left(\dfrac{x}{2}\right)\right) = \dfrac{\sqrt{4-x^2}}{2}$ for $-2 \leq x \leq 2$

\item $\cos\left(\arctan\left(3x\right)\right) = \dfrac{1}{\sqrt{1+9x^{2}}}$ for all $x$

\item $\sin(2\arcsin(7x)) = 14x \sqrt{1-49x^2}$ for $-\dfrac{1}{7} \leq x \leq \dfrac{1}{7}$

\item $\sin\left(2 \arcsin\left( \dfrac{x\sqrt{3}}{3} \right) \right) = \dfrac{2x\sqrt{3-x^2}}{3}$ for $-\sqrt{3} \leq x \leq \sqrt{3}$

\item $\cos(2 \arcsin(4x)) = 1 - 32x^2$ for $-\dfrac{1}{4} \leq x \leq \dfrac{1}{4}$

\item $\sec(\arctan(2x))\tan(\arctan(2x)) = 2x \sqrt{1+4x^2}$ for all $x$

\item $\sin \left( \arcsin(x) + \arccos(x) \right) = 1$ for $-1 \leq x \leq 1$

\item $\cos \left( \arcsin(x) + \arctan(x) \right) = \dfrac{\sqrt{1 - x^{2}} - x^{2}}{\sqrt{1 + x^{2}}}$ for $-1 \leq x \leq 1$

\item\footnote{The equivalence for $x = \pm 1$ can be verified independently of the derivation of the formula, but Calculus is required to fully understand what is happening at those $x$ values. You'll see what we mean when you work through the details of the identity for $\tan(2t).$ For now, we exclude $x = \pm 1$ from our answer.} $\tan \left( 2\arcsin(x) \right) = \dfrac{2x\sqrt{1 - x^{2}}}{1 - 2x^{2}}$ for $x$ in $\left(-1, -\dfrac{\sqrt{2}}{2}\right) \cup \left(-\dfrac{\sqrt{2}}{2}, \dfrac{\sqrt{2}}{2} \right) \cup \left(\dfrac{\sqrt{2}}{2}, 1\right)$

\item $\sin \left( \dfrac{1}{2}\arctan(x) \right) = \left\{ \begin{array}{rr} \sqrt{\dfrac{\sqrt{x^{2} + 1} - 1}{2\sqrt{x^{2} + 1}}} & \text{for $x \geq 0$} \\ [10pt] -\sqrt{\dfrac{\sqrt{x^{2} + 1} - 1}{2\sqrt{x^{2} + 1}}} & \text{for $x < 0$} \end{array}\right. $

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item If $\sin(\theta) = \dfrac{x}{2}$ for $-\dfrac{\pi}{2} < \theta < \dfrac{\pi}{2}$, then $\theta + \sin(2\theta) = \arcsin \left( \dfrac{x}{2} \right) + \dfrac{x\sqrt{4 - x^{2}}}{2}$

\item If $\tan(\theta) = \dfrac{x}{7}$ for $-\dfrac{\pi}{2} < \theta < \dfrac{\pi}{2}$, then $\dfrac{1}{2}\theta - \dfrac{1}{2}\sin(2\theta) = \dfrac{1}{2} \arctan \left( \dfrac{x}{7} \right) - \dfrac{7x}{x^{2} + 49}$

\item If $\sec(\theta) = \dfrac{x}{4}$ for $0 < \theta < \dfrac{\pi}{2}$, then $4\tan(\theta) - 4\theta = \sqrt{x^{2} - 16} - 4\mbox{arcsec} \left( \dfrac{x}{4} \right)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $x = \arcsin\left(\dfrac{7}{11}\right) + 2\pi k$ or $x = \pi - \arcsin\left(\dfrac{7}{11}\right) + 2\pi k$, in $[0, 2\pi)$, $x \approx 0.6898, \, 2.4518$

\item $x = \arccos\left(-\dfrac{2}{9}\right) + 2\pi k$ or $x = - \arccos\left(-\dfrac{2}{9}\right) + 2\pi k$, in $[0, 2\pi)$, $x \approx 1.7949, \, 4.4883$

\item $x = \pi + \arcsin(0.569) + 2\pi k$ or $x = 2\pi - \arcsin(0.569) + 2\pi k$, in $[0, 2\pi)$, $x \approx 3.7469, \, 5.6779$

\item $x = \arccos(0.117) + 2\pi k$ or $x = 2\pi - \arccos(0.117) + 2\pi k$, in $[0, 2\pi)$, $x \approx 1.4535, \, 4.8297$

\item $x = \arcsin(0.008) + 2\pi k$ or $x = \pi - \arcsin(0.008) + 2\pi k$, in $[0, 2\pi)$, $x \approx 0.0080, \, 3.1336$

\item $x = \arccos\left(\dfrac{359}{360}\right) + 2\pi k$ or $x = 2\pi - \arccos\left(\dfrac{359}{360}\right) + 2\pi k$, in $[0, 2\pi)$, $x \approx 0.0746, \, 6.2086$

\item $x = \arctan(117) + \pi k$, in $[0, 2\pi)$, $x \approx 1.56225, \, 4.70384$

\item $x = \arctan\left(-\dfrac{1}{12}\right) + \pi k$, in $[0, 2\pi)$, $x \approx 3.0585, \, 6.2000$

\item $x = \arccos\left(\dfrac{2}{3}\right) + 2\pi k$ or $x = 2\pi - \arccos\left(\dfrac{2}{3}\right) + 2\pi k$, in $[0, 2\pi)$, $x \approx 0.8411, \, 5.4422$

\item $x = \pi + \arcsin\left(\dfrac{17}{90}\right) + 2\pi k$ or $x = 2\pi - \arcsin\left(\dfrac{17}{90}\right) + 2\pi k$, in $[0, 2\pi)$, $x \approx 3.3316, \, 6.0932$

\item $x = \arctan\left(-\sqrt{10}\right) + \pi k$, in $[0, 2\pi)$, $x \approx 1.8771, \, 5.0187$

\item $x = \arcsin\left(\dfrac{3}{8}\right) + 2\pi k$ or $x = \pi - \arcsin\left(\dfrac{3}{8}\right) + 2\pi k$, in $[0, 2\pi)$, $x \approx 0.3844, \, 2.7572$

\item $x = \arccos\left(-\dfrac{7}{16}\right) + 2\pi k$ or $x = - \arccos\left(-\dfrac{7}{16}\right) + 2\pi k$, in $[0, 2\pi)$, $x \approx 2.0236, \, 4.2596$

\item $x = \arctan(0.03) + \pi k$, in $[0, 2\pi)$, $x \approx 0.0300, \, 3.1716$

\item $x = \arcsin(0.3502) + 2\pi k$ or $x = \pi - \arcsin(0.3502) + 2\pi k$, in $[0, 2\pi)$, $x \approx 0.3578, \,2.784$

\item $x = \pi + \arcsin(0.721) + 2\pi k$ or $x = 2\pi - \arcsin(0.721) + 2\pi k$, in $[0, 2\pi)$, $x \approx 3.9468, \, 5.4780$

\item $x = \arccos(0.9824) + 2\pi k$ or $x = 2\pi - \arccos(0.9824) + 2\pi k$, in $[0, 2\pi)$, $x \approx 0.1879, \, 6.0953$

\item $x = \arccos(-0.5637) + 2\pi k$ or $x = - \arccos(-0.5637) + 2\pi k$, in $[0, 2\pi)$, $x \approx 2.1697, \, 4.1135$

\item $x = \arctan(117) + \pi k$, in $[0, 2\pi)$, $x \approx 1.5622, \, 4.7038$

\item $x = \arctan(-0.6109) + \pi k$, in $[0, 2\pi)$, $x \approx 2.5932, \, 5.7348$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $36.87^{\circ}$ and $53.13^{\circ}$

\item $22.62^{\circ}$ and $67.38^{\circ}$

\item $32.52^{\circ}$ and $57.48^{\circ}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{5}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $68.9^{\circ}$

\item $7.7^{\circ}$

\item $51^{\circ}$

\item $19.5^{\circ}$

\item $41.81^{\circ}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = 5\sin(3x) + 12\cos(3x) = 13\sin\left(3x + \arcsin\left(\dfrac{12}{13}\right)\right) \approx 13\sin(3x + 1.1760)$

\item $f(x) = 3\cos(2x) + 4\sin(2x) = 5\sin\left(2x+\arcsin\left(\dfrac{3}{5}\right) \right) \approx 5\sin(2x+0.6435)$

\item $f(x) = \cos(x) - 3\sin(x) = \sqrt{10} \sin\left(x + \arccos\left(-\dfrac{3\sqrt{10}}{10} \right)\right) \approx \sqrt{10} \sin(x + 2.8198)$

\item $f(x) = 7\sin(10x) - 24\cos(10x) = 25\sin\left( 10x + \arcsin\left(-\dfrac{24}{25}\right)\right) \approx 25 \sin(10x-1.2870)$

\item $f(x) = -\cos(x) - 2\sqrt{2} \sin(x) = 3\sin\left(x+\pi + \arcsin\left(\dfrac{1}{3}\right)\right) \approx 3\sin(x+3.4814)$

\item $f(x) = 2\sin(x) - \cos(x) = \sqrt{5}\sin\left(x + \arcsin\left(-\dfrac{\sqrt{5}}{5}\right)\right) \approx \sqrt{5}\sin(x -0.4636)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\left[-\dfrac{1}{5}, \dfrac{1}{5}\right]$

\item $\left[-\dfrac{1}{3}, 1 \right]$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\left[-\dfrac{\sqrt{2}}{2}, \dfrac{\sqrt{2}}{2}\right]$

\item $(-\infty, -\sqrt{5}] \cup [-\sqrt{3}, \sqrt{3}] \cup [\sqrt{5}, \infty)$ \vphantom{$\left[ \dfrac{\sqrt{2}}{2}\right]$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $(-\infty, \infty)$

\item $(-\infty, -3) \cup (-3,3) \cup (3, \infty)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\left(\dfrac{1}{2}, \infty \right)$

\item $\left[\dfrac{1}{2}, \infty \right)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\left(-\infty, -\dfrac{1}{12}\right] \cup \left[\dfrac{1}{12}, \infty\right)$

\item $(-\infty, -6] \cup [-4, \infty)$ \vphantom{$\left[ -\dfrac{1}{12}\right]$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $(-\infty, -2] \cup [2, \infty)$

\item $[0, \infty)$

\end{enumerate}

\end{multicols}

\closegraphsfile

10.7: Trigonometric Equations and Inequalities

\subsection{Exercises}

In Exercises \ref{solvebasicfirst} - \ref{solvebasiclast}, find \underline{all} of the exact solutions of the equation and then list those solutions which are in the interval $[0, 2\pi)$.

\begin{multicols}{3}

\begin{enumerate}

\item $\sin \left( 5x \right) = 0$ \vphantom{$\dfrac{\sqrt{3}}{2}$} \label{solvebasicfirst}

\item $\cos \left( 3x \right) = \dfrac{1}{2}$ \vphantom{$\dfrac{\sqrt{3}}{2}$}

\item $\sin \left( -2x \right) = \dfrac{\sqrt{3}}{2}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\tan \left( 6x \right) = 1$

\item $\csc \left( 4x \right) = -1$

\item $\sec \left( 3x \right) = \sqrt{2}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cot \left( 2x \right) = -\dfrac{\sqrt{3}}{3}$

\item $\cos \left( 9x \right) = 9$ \vphantom{$\dfrac{\sqrt{3}}{2}$}

\item $\sin \left( \dfrac{x}{3} \right) = \dfrac{\sqrt{2}}{2}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos \left( x + \dfrac{5\pi}{6} \right) = 0$

\item $\sin \left( 2x - \dfrac{\pi}{3} \right) = -\dfrac{1}{2}$

\item $2\cos \left( x + \dfrac{7\pi}{4} \right) = \sqrt{3}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\csc(x) = 0$

\item $\tan \left( 2x - \pi \right) = 1$

\item $\tan^{2} \left( x \right) = 3$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sec^{2} \left( x \right) = \dfrac{4}{3}$

\item $\cos^{2} \left( x \right) = \dfrac{1}{2}$

\item $\sin^{2} \left( x \right) = \dfrac{3}{4}$ \label{solvebasiclast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

In Exercises \ref{solveidentfirst} - \ref{solveidentlast}, solve the equation, giving the exact solutions which lie in $[0, 2\pi)$

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sin \left( x \right) = \cos \left( x \right)$ \label{solveidentfirst}

\item $\sin \left( 2x \right) = \sin \left( x \right)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sin \left( 2x \right) = \cos \left( x \right)$

\item $\cos \left( 2x \right) = \sin \left( x \right)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos \left( 2x \right) = \cos \left( x \right)$

\item $\cos(2x) = 2 - 5\cos(x)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $3\cos(2x) + \cos(x) + 2 = 0$

\item $\cos(2x) = 5\sin(x) - 2$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $3\cos(2x) = \sin(x) + 2$

\item $2\sec^{2}(x) = 3 - \tan(x)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\tan^{2}(x) = 1-\sec(x)$

\item $\cot^{2}(x) = 3\csc(x) - 3$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sec(x) = 2\csc(x)$

\item $\cos(x)\csc(x)\cot(x) = 6-\cot^{2}(x)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sin(2x) = \tan(x)$

\item $\cot^{4}(x) = 4\csc^{2}(x) - 7$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos(2x) + \csc^{2}(x) = 0$

\item $\tan^{3} \left( x \right) = 3\tan \left( x \right)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\tan^{2} \left( x \right) = \dfrac{3}{2} \sec \left( x \right)$

\item $\cos^{3} \left( x \right) = -\cos \left( x \right)$ \vphantom{$\dfrac{3}{2}$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\tan (2x) - 2\cos(x) = 0$

\item $\csc^{3}(x) + \csc^{2}(x) = 4\csc(x) + 4$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\enlargethispage{.5in}

\vspace{-.1in}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $2\tan(x) = 1 - \tan^{2}(x)$

\item $\tan \left( x \right) = \sec \left( x \right)$ \label{solveidentlast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

In Exercises \ref{solvemoreidentfirst} - \ref{solvemoreidentlast}, solve the equation, giving the exact solutions which lie in $[0, 2\pi)$

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sin(6x) \cos(x) = -\cos(6x) \sin(x)$ \label{solvemoreidentfirst}

\item $\sin(3x)\cos(x) = \cos(3x) \sin(x)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos(2x)\cos(x) + \sin(2x)\sin(x) = 1$ \vphantom{$\dfrac{\sqrt{3}}{2}$}

\item \small $\cos(5x)\cos(3x) - \sin(5x)\sin(3x) = \dfrac{\sqrt{3}}{2}$ \normalsize

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

%Sinusoids

\item $\sin(x) + \cos(x) = 1$

\item $\sin(x) + \sqrt{3} \cos(x) = 1$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sqrt{2} \cos(x) - \sqrt{2} \sin(x) = 1$

\item $\sqrt{3} \sin(2x) + \cos(2x) = 1$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos(2x) - \sqrt{3} \sin(2x) = \sqrt{2}$

\item $3\sqrt{3}\sin(3x) - 3\cos(3x) = 3\sqrt{3}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos(3x) = \cos(5x)$

\item $\cos(4x) = \cos(2x)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sin(5x) = \sin(3x)$

\item $\cos(5x) = -\cos(2x)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sin(6x) + \sin(x) = 0$

\item $\tan(x) = \cos(x)$ \label{solvemoreidentlast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

In Exercises \ref{firstineqfirst} - \ref{firstineqlast}, solve the inequality. Express the exact answer in \underline{interval} notation, restricting your attention to $0 \leq x \leq 2\pi$.

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sin \left( x \right) \leq 0$ \label{firstineqfirst}

\item $\tan \left( x \right) \geq \sqrt{3}$

\item $\sec^{2} \left( x \right) \leq 4$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos^{2} \left( x \right) > \dfrac{1}{2}$

\item $\cos \left( 2x \right) \leq 0$ \vphantom{$\dfrac{1}{2}$}

\item $\sin \left( x + \dfrac{\pi}{3} \right) > \dfrac{1}{2}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cot^{2} \left( x \right) \geq \dfrac{1}{3}$

\item $2\cos(x) \geq 1$ \vphantom{$\dfrac{1}{2}$}

\item $\sin(5x) \geq 5$ \vphantom{$\dfrac{1}{2}$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos(3x) \leq 1$

\item $\sec(x) \leq \sqrt{2}$

\item $\cot(x) \leq 4$ \label{firstineqlast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

In Exercises \ref{secondineqefirst} - \ref{secondineqlast}, solve the inequality. Express the exact answer in \underline{interval} notation, restricting your attention to $-\pi \leq x \leq \pi$.

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\cos \left( x \right) > \dfrac{\sqrt{3}}{2}$ \label{secondineqefirst}

\item $\sin(x) > \dfrac{1}{3}$ \vphantom{$\dfrac{\sqrt{3}}{2}$}

\item $\sec \left( x \right) \leq 2$ \vphantom{$\dfrac{\sqrt{3}}{2}$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\sin^{2} \left( x \right) < \dfrac{3}{4}$

\item $\cot \left( x \right) \geq -1$ \vphantom{$\dfrac{1}{2}$}

\item $\cos(x) \geq \sin(x)$ \vphantom{$\dfrac{1}{2}$} \label{secondineqlast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

%\pagebreak

In Exercises \ref{thirdineqfirst} - \ref{thirdineqlast}, solve the inequality. Express the exact answer in \underline{interval} notation, restricting your attention to $-2\pi \leq x \leq 2\pi$.

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\csc \left( x \right) > 1$ \vphantom{$\dfrac{1}{2}$} \label{thirdineqfirst}

\item $\cos(x) \leq \dfrac{5}{3}$

\item $\cot(x) \geq 5$ \vphantom{$\dfrac{1}{2}$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\tan^{2} \left( x \right) \geq 1$

\item $\sin(2x) \geq \sin(x)$

\item $\cos(2x) \leq \sin(x)$ \label{thirdineqlast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

In Exercises \ref{domainfirst} - \ref{domainlast}, express the domain of the function using the extended interval notation. (See page \pageref{extendedinterval} in Section \ref{circularfunctionsbeyond} for details.)

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = \dfrac{1}{\cos(x) - 1}$ \vphantom{$\dfrac{\cos(x)}{\sin(x) + 1}$} \label{domainfirst}

\item $f(x) = \dfrac{\cos(x)}{\sin(x) + 1}$

\item $f(x) = \sqrt{\tan^{2}(x) - 1}$ \vphantom{$\dfrac{\cos(x)}{\sin(x) + 1}$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = \sqrt{2 - \sec(x)}$ \vphantom{$\dfrac{\cos(x)}{\sin(x) + 1}$}

\item $f(x) = \csc(2x)$ \vphantom{$\dfrac{\cos(x)}{\sin(x) + 1}$}

\item $f(x) = \dfrac{\sin(x)}{2 + \cos(x)}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{3}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $f(x) = 3\csc(x) + 4\sec(x)$

\item $f(x) = \ln\left( |\cos(x)| \right)$

\item $f(x) = \arcsin(\tan(x))$ \label{domainlast}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item With the help of your classmates, determine the number of solutions to $\sin(x) = \frac{1}{2}$ in $[0,2\pi)$. Then find the number of solutions to $\sin(2x) = \frac{1}{2}$, $\sin(3x) = \frac{1}{2}$ and $\sin(4x) = \frac{1}{2}$ in $[0,2\pi)$. A pattern should emerge. Explain how this pattern would help you solve equations like $\sin(11x) = \frac{1}{2}$. Now consider $\sin\left(\frac{x}{2}\right) = \frac{1}{2}$, $\sin\left(\frac{3x}{2}\right) = \frac{1}{2}$ and $\sin\left(\frac{5x}{2}\right) = \frac{1}{2}$. What do you find? Replace $\dfrac{1}{2}$ with $-1$ and repeat the whole exploration.

\end{enumerate}

\newpage

\subsection{Answers}

\begin{enumerate}

\item $x = \dfrac{\pi k}{5}; \; x = 0, \dfrac{\pi}{5}, \dfrac{2\pi}{5}, \dfrac{3\pi}{5}, \dfrac{4\pi}{5}, \pi, \dfrac{6\pi}{5}, \dfrac{7\pi}{5}, \dfrac{8\pi}{5}, \dfrac{9\pi}{5}$

\item $x = \dfrac{\pi}{9} + \dfrac{2\pi k}{3}$ or $x = \dfrac{5\pi}{9} + \dfrac{2\pi k}{3}; \; x = \dfrac{\pi}{9}, \dfrac{5\pi}{9}, \dfrac{7\pi}{9}, \dfrac{11\pi}{9}, \dfrac{13\pi}{9}, \dfrac{17\pi}{9}$

\item $x = \dfrac{2\pi}{3} + \pi k$ or $x = \dfrac{5\pi}{6} + \pi k; \; x = \dfrac{2\pi}{3}, \dfrac{5\pi}{6}, \dfrac{5\pi}{3}, \dfrac{11\pi}{6}$

\item $x = \dfrac{\pi}{24} + \dfrac{\pi k}{6}; \; x = \dfrac{\pi}{24}, \dfrac{5\pi}{24}, \dfrac{3\pi}{8}, \dfrac{13\pi}{24}, \dfrac{17\pi}{24}, \dfrac{7\pi}{8}, \dfrac{25\pi}{24}, \dfrac{29\pi}{24}, \dfrac{11\pi}{8}, \dfrac{37\pi}{24}, \dfrac{41\pi}{24}, \dfrac{15\pi}{8}$

\item $x = \dfrac{3\pi}{8} + \dfrac{\pi k}{2}; \; x = \dfrac{3\pi}{8}, \dfrac{7\pi}{8}, \dfrac{11\pi}{8}, \dfrac{15\pi}{8}$

\item $x = \dfrac{\pi}{12} + \dfrac{2\pi k}{3}$ or $x = \dfrac{7\pi}{12} + \dfrac{2\pi k}{3}; \; x = \dfrac{\pi}{12}, \dfrac{7\pi}{12}, \dfrac{3\pi}{4}, \dfrac{5\pi}{4}, \dfrac{17\pi}{12}, \dfrac{23\pi}{12}$

\item $x = \dfrac{\pi}{3} + \dfrac{\pi k}{2}; \; x = \dfrac{\pi}{3}, \dfrac{5\pi}{6}, \dfrac{4\pi}{3}, \dfrac{11\pi}{6}$

\item No solution

\item $x = \dfrac{3\pi}{4} + 6\pi k$ or $x = \dfrac{9\pi}{4} + 6\pi k; \; x = \dfrac{3\pi}{4}$

\item $x = -\dfrac{\pi}{3} + \pi k; \; x = \dfrac{2\pi}{3}, \dfrac{5\pi}{3}$

\item $x = \dfrac{3\pi}{4} + \pi k$ or $x = \dfrac{13\pi}{12} + \pi k; \; x = \dfrac{\pi}{12}, \dfrac{3\pi}{4}, \dfrac{13\pi}{12}, \dfrac{7\pi}{4}$

\item $x = -\dfrac{19\pi}{12} + 2\pi k$ or $x = \dfrac{\pi}{12} + 2\pi k; \; x = \dfrac{\pi}{12}, \dfrac{5\pi}{12}$

\item No solution

\item $x = \dfrac{5\pi}{8} + \dfrac{\pi k}{2}; \; x = \dfrac{\pi}{8}, \dfrac{5\pi}{8}, \dfrac{9\pi}{8}, \dfrac{13\pi}{8}$

\item $x = \dfrac{\pi}{3} + \pi k$ or $x = \dfrac{2\pi}{3} + \pi k; \; x = \dfrac{\pi}{3}, \dfrac{2\pi}{3}, \dfrac{4\pi}{3}, \dfrac{5\pi}{3}$

\item $x = \dfrac{\pi}{6} + \pi k$ or $x = \dfrac{5\pi}{6} + \pi k; \; x = \dfrac{\pi}{6}, \dfrac{5\pi}{6}, \dfrac{7\pi}{6}, \dfrac{11\pi}{6}$

\item $x = \dfrac{\pi}{4} + \dfrac{\pi k}{2}; \; x = \dfrac{\pi}{4}, \dfrac{3\pi}{4}, \dfrac{5\pi}{4}, \dfrac{7\pi}{4}$

\item $x = \dfrac{\pi}{3} + \pi k$ or $x = \dfrac{2\pi}{3} + \pi k; \; x = \dfrac{\pi}{3}, \dfrac{2\pi}{3}, \dfrac{4\pi}{3}, \dfrac{5\pi}{3}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $x = \dfrac{\pi}{4}, \dfrac{5\pi}{4}$

\item $x = 0, \dfrac{\pi}{3}, \pi, \dfrac{5\pi}{3}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $x = \dfrac{\pi}{6}, \dfrac{\pi}{2}, \dfrac{5\pi}{6}, \dfrac{3\pi}{2}$

\item $x = \dfrac{\pi}{6}, \dfrac{5\pi}{6}, \dfrac{3\pi}{2}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $x = 0, \dfrac{2\pi}{3}, \dfrac{4\pi}{3}$

\item $x=\dfrac{\pi}{3}, \dfrac{5\pi}{3}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $x = \dfrac{2\pi}{3}, \dfrac{4\pi}{3}, \arccos\left(\dfrac{1}{3}\right), 2\pi -\arccos\left(\dfrac{1}{3}\right) $

\item $x=\dfrac{\pi}{6}, \dfrac{5\pi}{6}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $x = \dfrac{7\pi}{6}, \dfrac{11\pi}{6}, \arcsin\left(\dfrac{1}{3}\right), \pi - \arcsin\left(\dfrac{1}{3}\right) $

\item $x=\dfrac{3\pi}{4}, \dfrac{7\pi}{4}, \arctan\left(\dfrac{1}{2}\right), \pi +\arctan\left(\dfrac{1}{2}\right) $

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $x=0, \dfrac{2\pi}{3}, \dfrac{4\pi}{3}$

\item $x=\dfrac{\pi}{6}, \dfrac{5\pi}{6}, \dfrac{\pi}{2}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $x=\arctan(2), \pi + \arctan(2)$ \vphantom{$\dfrac{7\pi}{6}$}

\item $x = \dfrac{\pi}{6}, \dfrac{7\pi}{6}, \dfrac{5\pi}{6}, \dfrac{11\pi}{6}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $x = 0, \pi, \dfrac{\pi}{4}, \dfrac{3\pi}{4}, \dfrac{5\pi}{4}, \dfrac{7\pi}{4}$

\item $x = \dfrac{\pi}{6}, \dfrac{\pi}{4}, \dfrac{3\pi}{4}, \dfrac{5\pi}{6}, \dfrac{7\pi}{6}, \dfrac{5\pi}{4}, \dfrac{7\pi}{4}, \dfrac{11\pi}{6}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $x = \dfrac{\pi}{2}, \dfrac{3\pi}{2}$

\item $x = 0, \dfrac{\pi}{3}, \dfrac{2\pi}{3}, \pi, \dfrac{4\pi}{3}, \dfrac{5\pi}{3}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $x = \dfrac{\pi}{3}, \dfrac{5\pi}{3}$

\item $x = \dfrac{\pi}{2}, \dfrac{3\pi}{2}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $x = \dfrac{\pi}{6}, \dfrac{\pi}{2}, \dfrac{5\pi}{6}, \dfrac{3\pi}{2}$

\item $x = \dfrac{\pi}{6}, \dfrac{5\pi}{6}, \dfrac{7\pi}{6}, \dfrac{3\pi}{2}, \dfrac{11\pi}{6}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $x = \dfrac{\pi}{8}, \dfrac{5\pi}{8}, \dfrac{9\pi}{8}, \dfrac{13\pi}{8}$

\item No solution \vphantom{$\dfrac{7\pi}{6}$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $x = 0, \dfrac{\pi}{7}, \dfrac{2\pi}{7}, \dfrac{3\pi}{7}, \dfrac{4\pi}{7}, \dfrac{5\pi}{7}, \dfrac{6\pi}{7}, \pi, \dfrac{8\pi}{7}, \dfrac{9\pi}{7}, \dfrac{10\pi}{7}, \dfrac{11\pi}{7}, \dfrac{12\pi}{7}, \dfrac{13\pi}{7}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $x=0, \dfrac{\pi}{2}, \pi, \dfrac{3\pi}{2}$

\item $x = 0$ \vphantom{$\dfrac{7\pi}{6}$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $x = \dfrac{\pi}{48}, \dfrac{11\pi}{48}, \dfrac{13\pi}{48}, \dfrac{23\pi}{48}, \dfrac{25\pi}{48}, \dfrac{35\pi}{48}, \dfrac{37\pi}{48}, \dfrac{47\pi}{48}, \dfrac{49\pi}{48}, \dfrac{59\pi}{48}, \dfrac{61\pi}{48}, \dfrac{71\pi}{48}, \dfrac{73\pi}{48}, \dfrac{83\pi}{48}, \dfrac{85\pi}{48}, \dfrac{95\pi}{48}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $x = 0, \dfrac{\pi}{2}$ \vphantom{$\dfrac{7\pi}{6}$}

\item $x = \dfrac{\pi}{2}, \dfrac{11\pi}{6}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $x = \dfrac{\pi}{12}, \dfrac{17\pi}{12}$

\item $x= 0, \pi, \dfrac{\pi}{3}, \dfrac{4\pi}{3}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $x = \dfrac{17 \pi}{24}, \dfrac{41 \pi}{24}, \dfrac{23\pi}{24}, \dfrac{47\pi}{24}$

\item $x = \dfrac{\pi}{6}, \dfrac{5\pi}{18}, \dfrac{5\pi}{6}, \dfrac{17\pi}{18}, \dfrac{3\pi}{2}, \dfrac{29\pi}{18}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $x = 0, \dfrac{\pi}{4}, \dfrac{\pi}{2}, \dfrac{3\pi}{4}, \pi, \dfrac{5\pi}{4}, \dfrac{3\pi}{2}, \dfrac{7\pi}{4}$

\item $x = 0, \dfrac{\pi}{3}, \dfrac{2\pi}{3}, \pi, \dfrac{4\pi}{3}, \dfrac{5\pi}{3}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $x = 0, \dfrac{\pi}{8}, \dfrac{3\pi}{8}, \dfrac{5\pi}{8}, \dfrac{7\pi}{8}, \pi, \dfrac{9\pi}{8}, \dfrac{11\pi}{8}, \dfrac{13\pi}{8}, \dfrac{15\pi}{8}$

\item $x = \dfrac{\pi}{7}, \dfrac{\pi}{3}, \dfrac{3\pi}{7}, \dfrac{5\pi}{7}, \pi, \dfrac{9\pi}{7}, \dfrac{11\pi}{7}, \dfrac{5\pi}{3}, \dfrac{13\pi}{7}$

\item $x = \dfrac{2\pi}{7}, \dfrac{4\pi}{7}, \dfrac{6\pi}{7}, \dfrac{8\pi}{7}, \dfrac{10\pi}{7}, \dfrac{12\pi}{7}, \dfrac{\pi}{5}, \dfrac{3\pi}{5}, \pi, \dfrac{7\pi}{5}, \dfrac{9\pi}{5}$

\item $x = \arcsin \left( \dfrac{-1 + \sqrt{5}}{2} \right) \approx 0.6662, \pi - \arcsin \left( \dfrac{-1 + \sqrt{5}}{2} \right) \approx 2.4754$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\left[ \pi, 2\pi \right]$ \vphantom{$\left[ \dfrac{7\pi}{6} \right]$}

\item $\left[ \dfrac{\pi}{3}, \dfrac{\pi}{2} \right) \cup \left[ \dfrac{4\pi}{3}, \dfrac{3\pi}{2} \right)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\left[ 0, \dfrac{\pi}{3} \right] \cup \left[ \dfrac{2\pi}{3}, \dfrac{4\pi}{3} \right] \cup \left[ \dfrac{5\pi}{3}, 2\pi \right]$

\item $\left[ 0, \dfrac{\pi}{4} \right) \cup \left( \dfrac{3\pi}{4}, \dfrac{5\pi}{4} \right) \cup \left( \dfrac{7\pi}{4}, 2\pi \right]$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\left[ \dfrac{\pi}{4}, \dfrac{3\pi}{4} \right] \cup \left[ \dfrac{5\pi}{4}, \dfrac{7\pi}{4} \right]$

\item $\left[ 0, \dfrac{\pi}{2} \right) \cup \left( \dfrac{11\pi}{6}, 2\pi \right]$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \small $\left( 0, \dfrac{\pi}{3} \right] \cup \left[ \dfrac{2\pi}{3}, \pi \right) \cup \left( \pi, \dfrac{4\pi}{3} \right] \cup \left[ \dfrac{5\pi}{3}, 2\pi \right)$ \normalsize

\item $\left[0, \dfrac{\pi}{3}\right] \cup \left[\dfrac{5\pi}{3}, 2\pi\right]$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item No solution

\item $[0, 2\pi]$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\left[0, \dfrac{\pi}{4} \right] \cup \left(\dfrac{\pi}{2}, \dfrac{3\pi}{2}\right) \cup \left[\dfrac{7\pi}{4}, 2\pi\right]$

\item $\left[\text{arccot}(4), \pi \right) \cup \left[ \pi + \text{arccot}(4), 2\pi\right)$ \vphantom{$\left[ \dfrac{7\pi}{6} \right]$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\left( -\dfrac{\pi}{6}, \dfrac{\pi}{6} \right)$ \vphantom{$\left( \dfrac{7\pi}{6} \right)$}

\item $\left( \arcsin\left(\dfrac{1}{3}\right), \pi - \arcsin\left(\dfrac{1}{3}\right) \right)$ \vphantom{$\left( \dfrac{7\pi}{6} \right)$}

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\left[ -\pi, -\dfrac{\pi}{2} \right) \cup \left[ -\dfrac{\pi}{3}, \dfrac{\pi}{3} \right] \cup \left( \dfrac{\pi}{2}, \pi \right]$ \vphantom{$\left[ \dfrac{7\pi}{6} \right]$}

\item $\left( -\dfrac{2\pi}{3}, -\dfrac{\pi}{3} \right) \cup \left( \dfrac{\pi}{3}, \dfrac{2\pi}{3} \right)$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\left( -\pi, -\dfrac{\pi}{4} \right] \cup \left( 0, \dfrac{3\pi}{4} \right]$

\item $\left[ -\dfrac{3\pi}{4}, \dfrac{\pi}{4} \right]$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item \small $\left( -2\pi, -\dfrac{3\pi}{2} \right) \cup \left( -\dfrac{3\pi}{2}, -\pi \right) \cup \left( 0, \dfrac{\pi}{2} \right) \cup \left( \dfrac{\pi}{2}, \pi \right)$ \normalsize

\item $[-2\pi, 2\pi]$ \vphantom{$\left[ \dfrac{7\pi}{6} \right]$}

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\end{enumerate}

\end{multicols}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\left(-2\pi, \text{arccot}(5) - 2\pi\right] \cup \left(-\pi, \text{arccot}(5) - \pi\right] \cup \left(0, \text{arccot}(5)\right] \cup \left(\pi, \pi + \text{arccot}(5)\right]$

\item \scriptsize $\left[ -\dfrac{7\pi}{4}, -\dfrac{3\pi}{2} \right) \cup \left( -\dfrac{3\pi}{2}, -\dfrac{5\pi}{4} \right] \cup \left[ -\dfrac{3\pi}{4}, -\dfrac{\pi}{2} \right) \cup \left( -\dfrac{\pi}{2}, -\dfrac{\pi}{4} \right] \cup \left[ \dfrac{\pi}{4}, \dfrac{\pi}{2} \right) \cup \left( \dfrac{\pi}{2}, \dfrac{3\pi}{4} \right] \cup \left[ \dfrac{5\pi}{4}, \dfrac{3\pi}{2} \right) \cup \left( \dfrac{3\pi}{2}, \dfrac{7\pi}{4} \right]$ \normalsize

\item $\left[ -2\pi, -\dfrac{5\pi}{3} \right] \cup \left[ -\pi, -\dfrac{\pi}{3} \right] \cup \left[ 0, \dfrac{\pi}{3} \right] \cup \left[ \pi, \dfrac{5\pi}{3} \right]$

\item $\left[ -\dfrac{11\pi}{6}, -\dfrac{7\pi}{6} \right] \cup \left[ \dfrac{\pi}{6}, \dfrac{5\pi}{6} \right] \cup, \left\{ -\dfrac{\pi}{2}, \dfrac{3\pi}{2} \right\}$

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\end{enumerate}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\displaystyle \bigcup_{k=-\infty}^{\infty} \left( 2k\pi, (2k+2)\pi \right)$

\item $\displaystyle \bigcup_{k=-\infty}^{\infty} \left( \dfrac{(4k - 1)\pi}{2}, \dfrac{(4k + 3)\pi}{2} \right)$

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\end{enumerate}

\end{multicols}

\begin{enumerate}

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\item $\displaystyle \bigcup_{k=-\infty}^{\infty} \left\{ \left[ \dfrac{(4k + 1)\pi}{4}, \dfrac{(2k + 1)\pi}{2} \right) \cup \left( \dfrac{(2k + 1)\pi}{2}, \dfrac{(4k + 3)\pi}{4} \right] \right\}$

\item $\displaystyle \bigcup_{k=-\infty}^{\infty} \left\{ \left[ \dfrac{(6k - 1)\pi}{3}, \dfrac{(6k + 1)\pi}{3} \right] \cup \left( \dfrac{(4k + 1)\pi}{2}, \dfrac{(4k + 3)\pi}{2} \right) \right\}$

\setcounter{HW}{\value{enumi}}

\end{enumerate}

\begin{multicols}{2}

\begin{enumerate}

\setcounter{enumi}{\value{HW}}

\item $\displaystyle \bigcup_{k=-\infty}^{\infty} \left( \dfrac{k\pi}{2}, \dfrac{(k+1)\pi}{2} \right)$

\item $(-\infty, \infty)$ \vphantom{$\displaystyle \bigcup_{k=-\infty}^{\infty}$}

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\end{enumerate}

\end{multicols}

\begin{multicols}{2}

\begin{enumerate}

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\item $\displaystyle \bigcup_{k=-\infty}^{\infty} \left( \dfrac{k\pi}{2}, \dfrac{(k+1)\pi}{2} \right)$

\item $\displaystyle \bigcup_{k=-\infty}^{\infty} \left( \dfrac{(2k - 1)\pi}{2}, \dfrac{(2k+1)\pi}{2} \right)$

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\end{enumerate}

\end{multicols}

\begin{enumerate}

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\item $\displaystyle \bigcup_{k=-\infty}^{\infty} \left[ \dfrac{(4k - 1)\pi}{4}, \dfrac{(4k+1)\pi}{4} \right]$

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\end{enumerate}

\closegraphsfile