Preparing for Calculus

P.7 Assess Your Understanding
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Concepts and Vocabulary

$y=\sin ^{-1}x$ if and only if $x=$ _____, where $-1\leq x\leq 1$ and $-\dfrac{\pi }{2}\leq y\leq \dfrac{\pi }{2}$ .

$\sin y$

True or False $\sin ^{-1} (\sin x) =x,$ where $-1\leq x$ $\leq 1$ .

False

True or False The domain of $y =\sin ^{-1}x$ is $-\dfrac{\pi }{2}\leq x\leq \dfrac{\pi }{2}$ .

False

True or False $\sin (\sin ^{-1}0) =0$ .

True

True or False $y =\tan ^{-1}x$ means $x=\tan y$ , where $-\infty <x<\infty$ and $-\dfrac{\pi }{2}\leq y\leq \dfrac{\pi }{2}$ .

False

True or False The domain of the inverse tangent function is the set of all real numbers.

True

True or False $\sec ^{-1}0.5$ is not defined.

True

True or False Trigonometric equations can have multiple solutions.

True

Practice Problems

In Problems 9–20, find the exact value of each expression.

$\sin ^{ - 1} \left( {\frac{{\sqrt 2 }}{2}} \right)$

$\frac{\pi}{4}$

$\sin ^{-1}\left( -\dfrac{1}{2}\right)$

$\sec^{-1}2$

$\frac{\pi}{3}$

$\tan^{-1}\sqrt{3}$

$\tan ^{-1}( -1)$

$-\frac{\pi}{4}$

$\sec ^{-1}\dfrac{2\sqrt{3}}{3}$

$\tan ^{-1}\left( \tan \dfrac{\pi }{4}\right)$

$\frac{\pi}{4}$

$\tan ^{-1} ( \sin 0)$

$\sin \left( \sin ^{-1}\dfrac{3}{5}\right)$

$\frac{3}{5}$

$\tan [ \sec ^{-1} ( -3) ]$

$\sin ^{-1}\left( \sin \dfrac{4\pi }{5}\right)$

$\frac{\pi}{5}$

$\sec ( \sec ^{-}{}^{1}3)$

Write $\cos ( \sin ^{-1}u)$ as an algebraic expression containing $u$ , where $\vert u \vert \leq 1$ .

$\sqrt{1-u^2}$

Write $\tan ( \sin ^{-1}u)$ as an algebraic expression containing $u$ , where $\vert u \vert \leq 1$ .

Write $\sec ( \tan ^{-1}u)$ as an algebraic expression containing $u$ .

$\sqrt{1+u^2}$

Show that $y =\sin ^{-1}x$ is an odd function. That is, show $\sin ^{-1} ( -x) =-\sin ^{-1}x$ .

Show that $y =\tan ^{-1}x$ is an odd function. That is, show $\tan ^{-1} ( -x) =-\tan ^{-1}x$ .

See Student Solutions Manual.

Given that $x=\sin ^{-1}\dfrac{1}{2}$ , find $\cos x$ , $\tan x$ , $\cot x$ , $\sec x$ , and $\csc x$ .

In Problems 27–44, solve each equation on the interval $0\leq \theta <2\pi$ .

$\tan \theta =-\dfrac{\sqrt{3}}{3}$

$\left\{\frac{5\pi}{6}, \frac{11\pi}{6}\right\}$

$\sec \dfrac{3\theta }{2}=-2$

$2\sin \theta +3=2$

$\left\{\frac{7\pi}{6}, \frac{11\pi}{6}\right\}$

$1-\cos \theta =\dfrac{1}{2}$

$\sin ( 3\theta ) =-1$

$\left\{\frac{\pi}{2}, \frac{7\pi}{6}, \frac{11\pi}{6}\right\}$

$\cos ( 2\theta ) =\dfrac{1}{2}$

$4\cos ^{2}\theta =1$

$\left\{\frac{\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3}, \frac{5\pi}{3}\right\}$

$2\sin ^{2}\theta -1=0$

$2\sin ^{2}\theta -5\sin \theta +3=0$

$\left\{\frac{\pi}{2}\right\}$

$2\cos ^{2}\theta -7\cos \theta -4=0$

$1+\sin \theta =2\cos ^{2}\theta$

$\left\{\frac{\pi}{6}, \frac{5\pi}{6}, \frac{3\pi}{2}\right\}$

$\sec ^{2}\theta +\tan \theta =0$

$\sin \theta +\cos \theta = 0$

$\left\{\frac{3\pi}{4}, \frac{7\pi}{4}\right\}$

$\tan \theta =\cot \theta$

$\cos (2\theta ) +6\sin ^{2}\theta =4$

$\left\{\frac{\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3}, \frac{5\pi}{3}\right\}$

$\cos ( 2\theta ) =\cos \theta$

$\mathbf{\sin }( 2\theta ) +\sin ( 4\theta ) =0$

$\left\{0, \frac{\pi}{3}, \frac{\pi}{2}, \frac{2\pi}{3}, \pi, \frac{4\pi}{3},\frac{3\pi}{2}, \frac{5\pi}{3}\right\}$

$\cos ( 4\theta) -\cos ( 6\theta ) =0$

In Problems 45–48, use a calculator to solve each equation on the interval $0\leq \theta <2\pi$ . Round answers to three decimal places.

$\tan \theta =5$

$\{1.373, 4.515\}$

$\cos \theta =0.6$

$2+3\sin \theta =0$

$\{3.871, 5.553\}$

$4+\sec \theta =0$

1. On the same set of axes, graph $f( x) = 3\sin ( 2x) +2$ and $g( x) =\dfrac{7}{2}$ on the interval $[ 0,\pi ]$ .
2. Solve $f( x) = g(x)$ on the interval $[ 0,\pi ]$ , and label the points of intersection on the graph drawn in (a).
3. Shade the region bounded by $f( x) =3\sin (2x) + 2$ and $g(x) = \dfrac{7}{2}$ between the points found in (b) on the graph drawn in (a).
4. Solve $f(x) > g(x)$ on the interval $[ 0,\pi ]$ .

$({\rm a \, {\rm and }\, \rm c})$
$\left(\frac{\pi}{12}, \frac{7}{2}\right)$ , $\left(\frac{5\pi}{12}, \frac{7}{2}\right)$
(c) See (a).
$\left\{x \left| \frac{\pi}{12} < x < \frac{5\pi}{12}\right\}\right.$

1. On the same set of axes, graph $f( x) =-4 \cos x$ and $g( x) =2\cos x+3$ on the interval $[ 0,2\pi ]$ .
2. Solve $f( x) =g( x)$ on the interval $[ 0,2\pi ]$ , and label the points of intersection on the graph drawn in (a).
3. Shade the region bounded by $f( x) = -4\cos x$ and $g( x) =2\cos x+3$ between the points found in (b) on the graph drawn in (a).
4. Solve $f(x) > g(x)$ on the interval $[ 0,2\pi ]$ .

P.7 Assess Your UnderstandingPrinted Page 56

P.7 Assess Your Understanding
Printed Page 56