Question 5.5.8: Evaluating compositions of functions Find the exact value of...
Evaluating compositions of functions
Find the exact value of each composition without using a table or a calculator.
a. \sin \left(\tan ^{-1}(0)\right) b. arcsin(cos(π/6)) c. \tan \left(\sec ^{-1}(\sqrt{2})\right) d. arcsin(sin(4π/3))
a. Since \tan (0)=0, \tan ^{-1}(0)=0. Therefore,
\sin \left(\tan ^{-1}(0)\right)=\sin (0)=0.
b. Since \cos (\pi / 6)=\sqrt{3} / 2, we have
\arcsin \left(\cos \left(\frac{\pi}{6}\right)\right)=\arcsin \left(\frac{\sqrt{3}}{2}\right)=\frac{\pi}{3}.
c. To find \sec ^{-1}(\sqrt{2}), we use the identity \sec ^{-1}(x)=\cos ^{-1}(1 / x). Since \cos (\pi / 4)=1 / \sqrt{2}, we have \cos ^{-1}(1 / \sqrt{2})=\pi / 4 and \sec ^{-1}(\sqrt{2})=\pi / 4. Therefore,
\tan \left(\sec ^{-1}(\sqrt{2})\right)=\tan \left(\frac{\pi}{4}\right)=1You can check parts (a), (b), and (c) with a graphing calculator, as shown in Fig. 5.80. Note that \pi / 3 \approx 1.047
d. Since \sin (4 \pi / 3)=-\sin (\pi / 3) we have \sin (4 \pi / 3)=-\sqrt{3} / 2. Now the angle in the interval [-π/2, π/2] whose sine is -\sqrt{3} / 2 is -π/3. So
\arcsin (\sin (4 \pi / 3))=\arcsin (-\sqrt{3} / 2)=-\pi / 3Note that \arcsin (\sin (4 \pi / 3)) \neq 4 \pi / 3.
