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19.2: Exercises

Last updated

May 2, 2022
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- 19.1: The functions of arcsin, arccos, and arctan
- 20: Trigonometric Equations

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 $\PageIndex{1}$

Graph the function with the calculator. Use both radian and degree mode to display your graph. Zoom to an appropriate window for each mode to display a graph which includes the main features of the graph.

$y=\sin^{-1}(x)$
$y=\cos^{-1}(x)$
$y=\tan^{-1}(x)$

Answer

$clipboard_e4c73a7e5ee509f3cac908fdd712db817.png$ $\begin{aligned}&-2 \leq x \leq 2 \\&-2 \leq y \leq 2\end{aligned}$
$clipboard_ec15b01d8fc10b8828825274f03238602.png$ $\begin{aligned}&-2 \leq x \leq 2 \\&-1 \leq y \leq 4\end{aligned}$
$clipboard_e8d1a15881b5138fab8bbb49f7cdde065.png$ $\begin{aligned}-10 & \leq x \leq 10 \\-2 & \leq y \leq 2\end{aligned}$

Exercise $\PageIndex{2}$

Find the exact value of the inverse trigonometric function.

$\tan^{-1}(\sqrt{3})$
$\sin^{-1}\left(\dfrac{1}{2}\right)$
$\cos^{-1}\left(\dfrac{1}{2}\right)$
$\tan^{-1}(0)$
$\cos^{-1}\left(\dfrac{\sqrt{2}}{2}\right)$
$\cos^{-1}\left(-\dfrac{\sqrt{2}}{2}\right)$
$\sin^{-1}(-1)$
$\tan^{-1}(-\sqrt{3})$
$\cos^{-1}\left(-\dfrac{\sqrt{3}}{2}\right)$
$\sin^{-1}\left(-\dfrac{\sqrt{2}}{2}\right)$
$\sin^{-1}\left(-\dfrac{\sqrt{3}}{2}\right)$
$\tan^{-1}\left(-\dfrac{1}{\sqrt{3}}\right)$

Answer

$\dfrac{\pi}{3}$
$\dfrac{\pi}{6}$
$\dfrac{\pi}{3}$
$0$
$\dfrac{\pi}{4}$
$\dfrac{3\pi}{4}$
$-\dfrac{\pi}{2}$
$-\dfrac{\pi}{3}$
$\dfrac{5 \pi}{6}$
$-\dfrac{\pi}{4}$
$-\dfrac{\pi}{3}$
$-\dfrac{\pi}{6}$

Exercise $\PageIndex{3}$

Find the inverse trigonometric function value using the calculator. Approximate your answer to the nearest hundredth.

For parts (a)-(f), write your answer in radian mode.

$\cos^{-1}(0.2)$
$\sin^{-1}(-0.75)$
$\cos^{-1}\left(\dfrac{1}{3}\right)$
$\tan^{-1}(100,000)$
$\tan^{-1}(-2)$
$\cos^{-1}(-2)$

For parts (g)-(l), write your answer in degree mode.

$\cos^{-1}(0.68)$
$\tan^{-1}(-1)$
$\sin^{-1}\left(\dfrac{\sqrt{2}+\sqrt{6}}{4}\right)$
$\tan^{-1}(100,000)$
$\cos^{-1}\left(\dfrac{\sqrt{2-\sqrt{2}}}{2}\right)$
$\tan^{-1}(2+\sqrt{3}-\sqrt{6}-\sqrt{2})$

Answer

$1.37$
$−0.85$
$1.23$
$1.57$
$−1.11$
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$47.16^{\circ}$
$-45^{\circ}$
$75^{\circ}$
$90.00^{\circ}$
$67.5^{\circ}$
$-7.5^{\circ}$

Exercise 19.2.1\PageIndex{1}

Exercise 19.2.2\PageIndex{2}

Exercise 19.2.3\PageIndex{3}

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Exercise $\PageIndex{1}$

Exercise $\PageIndex{2}$

Exercise $\PageIndex{3}$