27.4: Review of trigonometric functions

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Page ID: 68485

Thomas Tradler and Holly Carley
CUNY New York City College of Technology via New York City College of Technology at CUNY Academic Works

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Exercise $\PageIndex{1}$

Fill in all the trigonometric function values in the table below.

\[\begin{array}{c||c|c|c|c|c}
& 0 & \dfrac{\pi}{6} & \dfrac{\pi}{4} & \dfrac{\pi}{3} & \dfrac{\pi}{2} \\
\hline \hline \sin (x) & & & & & \\
\hline \cos (x) & & & & & \\
\hline \tan (x) & & & & &
\end{array} \nonumber \]

Answer: $\begin{array}{c||c|c|c|c|c}
x & 0=0^{\circ} & \dfrac{\pi}{6}=30^{\circ} & \dfrac{\pi}{4}=45^{\circ} & \dfrac{\pi}{3}=60^{\circ} & \dfrac{\pi}{2}=90^{\circ} \\
\hline \hline \sin (x) & 0 & \dfrac{1}{2} & \dfrac{\sqrt{2}}{2} & \dfrac{\sqrt{3}}{2} & 1 \\
\hline \cos (x) & 1 & \dfrac{\sqrt{3}}{2} & \dfrac{\sqrt{2}}{2} & \dfrac{1}{2} & 0 \\
\hline \tan (x) & 0 & \dfrac{\sqrt{3}}{3} & 1 & \sqrt{3} & \text { undef. }
\end{array}$

Exercise $\PageIndex{2}$

Find the exact values of

$\cos\left(-\dfrac{\pi}{6}\right)$
$\sin\left(-\dfrac{\pi}{4}\right)$
$\tan\left(-\dfrac{\pi}{3}\right)$

Answer

$\dfrac{\sqrt{3}}{2}$
$\dfrac{-\sqrt{2}}{2}$
$-\sqrt{3}$

Exercise $\PageIndex{3}$

Find the exact value of the trigonometric function.

$\sin\left(\dfrac{5\pi}{4}\right)$
$\cos\left(\dfrac{11\pi}{6}\right)$

[Hint: Use the special $45^\circ-45^\circ-90^\circ$ or $30^\circ-60^\circ-90^\circ$ triangles to find the solution.]

Answer

$\dfrac{-\sqrt{2}}{2}$
$\dfrac{\sqrt{3}}{2}$

Exercise $\PageIndex{4}$

Find the amplitude, period, and the phase shift of the given function. Draw the graph over a one-period interval. Label all maxima, minima and intercepts.

$y=3 \cos\left(4 x-\pi\right)$
$y=-5\sin\left(x+\dfrac{\pi}{2}\right)$

Answer

amplitude $3$, period $\dfrac{\pi}{2}$, phase-shift $\dfrac{\pi}{4}$ $clipboard_eae7381b7075b464c7074552724f5f1ed.png$
amplitude $5$, period $2\pi $, phase-shift $\dfrac{-\pi}{2}$ $clipboard_e24e4d7c331858767be9f4c23ff7a2c75.png$

Exercise $\PageIndex{5}$

Find the exact trigonometric function value.

$\cos\left(\dfrac{\pi}{12}\right)$ [Hint: Use the addition and subtraction of angles formulas.]
$\cos\left(\dfrac{3\pi}{8}\right)$ [Hint: Use the half-angles formulas.]

Answer

$\dfrac{\sqrt{2}+\sqrt{6}}{4}$
$\dfrac{\sqrt{2-\sqrt{2}}}{2}$

Exercise $\PageIndex{6}$

Let $\sin(\alpha)=-\dfrac{4}{5}$ and let $\alpha$ be in quadrant III. Find $\sin(2\alpha)$, $\cos(2\alpha)$, and $\tan(2\alpha)$.

Answer: $\sin (2 \alpha)=\dfrac{24}{25}, \cos (2 \alpha)=\dfrac{-7}{25}, \tan (2 \alpha)=\dfrac{-24}{7}$

Exercise $\PageIndex{7}$

Find the exact value of:

$\sin^{-1}\left(-\dfrac{1}{2}\right)$
$\cos^{-1}\left(-\dfrac{\sqrt{3}}{2}\right)$
$\tan^{-1}\left(-\dfrac{\sqrt{3}}{3}\right)$

Answer

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

Exercise $\PageIndex{8}$

Solve for $x$: $2\sin(x)+\sqrt{3}=0$

Answer: $x=(-1)^{n+1} \dfrac{\pi}{3}+n \pi$, where $n=0, \pm 1, \ldots $

Exercise $\PageIndex{9}$

Solve for $x$: $\tan^2(x)-1=0$

Answer: $x=\pm \dfrac{\pi}{4}+n \pi $ where $n=0, \pm 1, \ldots $

Exercise $\PageIndex{10}$

Solve for $x$.

$2\cos^2(x)-1=0$
$2\sin^2(x)+15\sin(x)+7=0$

Answer

$x=\pm \dfrac{\pi}{4}+2 n \pi$ or $x=\pm \dfrac{3 \pi}{4}+2 n \pi$ where $n=0, \pm 1, \ldots$
$(-1)^{n+1} \dfrac{\pi}{6}+n \pi$ where $n=0, \pm 1, \ldots$

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