10.E: Foundations of Trigonometry (Exercises)
- Page ID
- 4766
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}
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\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}
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\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}
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\item $330^{\circ}$ \label{orientedanglefirst}
\item $-135^{\circ}$
\item $120^{\circ}$
\item $405^{\circ}$
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\end{enumerate}
\end{multicols}
\begin{multicols}{4}
\begin{enumerate}
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\item $-270^{\circ}$ \vphantom{$\dfrac{11\pi}{6}$}
\item $\dfrac{5\pi}{6}$
\item $-\dfrac{11\pi}{3}$
\item $\dfrac{5\pi}{4}$
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\end{enumerate}
\end{multicols}
\begin{multicols}{4}
\begin{enumerate}
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\item $\dfrac{3\pi}{4}$
\item $-\dfrac{\pi}{3}$ \vphantom{$\dfrac{11\pi}{6}$}
\item $\dfrac{7\pi}{2}$
\item $\dfrac{\pi}{4}$ \vphantom{$\dfrac{11\pi}{6}$}
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\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{$\dfrac{11\pi}{6}$}
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\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}
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\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 $-270^{\circ}$
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\end{enumerate}
\end{multicols}
\begin{multicols}{4}
\begin{enumerate}
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\item $-315^{\circ}$
\item $150^{\circ}$
\item $45^{\circ}$
\item $-225^{\circ}$ \label{degreetoradianlast}
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\end{enumerate}
\end{multicols}
In Exercises \ref{radiantodegreefirst} - \ref{radiantodegreelast}, convert the angle from radian measure into degree measure.
\begin{multicols}{4}
\begin{enumerate}
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\item $\pi$ \vphantom{$\dfrac{11\pi}{6}$} \label{radiantodegreefirst}
\item $-\dfrac{2\pi}{3}$
\item $\dfrac{7\pi}{6}$
\item $\dfrac{11\pi}{6}$
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\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}
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\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}
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\item $t=\frac{5 \pi}{6}$ \label{orientedarcfirst}
\item $t=-\pi$
\item $t = 6$
\item $t = -2$
\item $t = 12$ \label{orientedarclast}
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\end{enumerate}
\end{multicols}
\begin{enumerate}
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\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{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{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{$\dfrac{5\pi}{4}$}
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\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}
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\end{enumerate}
\end{multicols}
\begin{enumerate}
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\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 $179^{\circ} 59' 56''$
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\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 $237.979^{\circ}$
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\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
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\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
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\begin{enumerate}
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\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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\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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\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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\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 $\dfrac{\sin(\alpha+\beta)}{\sin(\alpha-\beta)} = \dfrac{1+\cot(\alpha) \tan(\beta)}{1 - \cot(\alpha) \tan(\beta)}$
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\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}
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\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}
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\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})
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\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})
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\end{enumerate}
\end{multicols}
\begin{multicols}{2}
\begin{enumerate}
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\item $\sin\left( \dfrac{\pi}{12} \right)$ (compare with Exercise \ref{sinpi12})
\item $\cos \left( \dfrac{\pi}{8} \right)$
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\end{enumerate}
\end{multicols}
\begin{multicols}{2}
\begin{enumerate}
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\item $\sin \left( \dfrac{5\pi}{8} \right)$
\item $\tan \left( \dfrac{7\pi}{8} \right)$ \label{idenhalfanglelast}
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\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 $0 < \theta < \dfrac{\pi}{2}$
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\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 $\dfrac{\pi}{2} < \theta < \pi$
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\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 $\pi < \theta < \dfrac{3\pi}{2}$
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\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}
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\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 $(\cos(\theta) - \sin(\theta))^2 = 1 - \sin(2\theta)$
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\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 $\csc(2\theta) = \dfrac{\cot(\theta) + \tan(\theta)}{2}$
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\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 $8 \cos^{4}(\theta) = \cos(4\theta) + 4\cos(2\theta)+3$
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\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 $\sin(4\theta) = 4\sin(\theta)\cos^{3}(\theta) - 4\sin^{3}(\theta)\cos(\theta)$
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\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}
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\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 $\sin(9\theta)\cos(\theta)$
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\end{enumerate}
\end{multicols}
\begin{multicols}{3}
\begin{enumerate}
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\item $\cos(2\theta) \cos(6\theta)$
\item $\sin(3\theta) \sin(2\theta)$
\item $\cos(\theta) \sin(3\theta)$ \label{prodsumlast}
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\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 $\cos(5\theta) - \cos(6\theta)$
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\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}
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\end{enumerate}
\end{multicols}
\begin{enumerate}
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\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}
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\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}
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\item No solution
\item $[0, 2\pi]$
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\end{enumerate}
\end{multicols}
\begin{multicols}{2}
\begin{enumerate}
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\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{$\left[ \dfrac{7\pi}{6} \right]$}
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\end{enumerate}
\end{multicols}
\begin{multicols}{2}
\begin{enumerate}
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\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{$\left( \dfrac{7\pi}{6} \right)$}
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\end{enumerate}
\end{multicols}
\begin{multicols}{2}
\begin{enumerate}
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\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 $\left( -\dfrac{2\pi}{3}, -\dfrac{\pi}{3} \right) \cup \left( \dfrac{\pi}{3}, \dfrac{2\pi}{3} \right)$
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\end{enumerate}
\end{multicols}
\begin{multicols}{2}
\begin{enumerate}
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\item $\left( -\pi, -\dfrac{\pi}{4} \right] \cup \left( 0, \dfrac{3\pi}{4} \right]$
\item $\left[ -\dfrac{3\pi}{4}, \dfrac{\pi}{4} \right]$
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\end{enumerate}
\end{multicols}
\begin{multicols}{2}
\begin{enumerate}
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\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}
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\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}
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\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 $\displaystyle \bigcup_{k=-\infty}^{\infty} \left\{ \left[ \dfrac{(6k - 1)\pi}{3}, \dfrac{(6k + 1)\pi}{3} \right] \cup \left( \dfrac{(4k + 1)\pi}{2}, \dfrac{(4k + 3)\pi}{2} \right) \right\}$
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\end{enumerate}
\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 $(-\infty, \infty)$ \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