Precalculus (10th Edition)

Published by Pearson
ISBN 10: 0-32197-907-9
ISBN 13: 978-0-32197-907-0

Chapter 7 - Analytic Trigonometry - 7.3 Trigonometric Equations - 7.3 Assess Your Understanding - Page 466: 57


$0.73, 2.41$.

Work Step by Step

The domain of $\sin^{-1}{x}$ is the range of $\sin{x}$, which is $[−1, 1] $. The range of $\sin^{-1}{x}$ is a specified domain of $\sin{x}$, i.e. $\left[− \frac{\pi}{2} , \frac{\pi}{2} \right]$. This equation is the same as: $\sin(\theta)=\frac{2}{3}$ because $3\sin(\theta)-2=0\\3\sin(\theta)=2\\\sin(\theta)=\frac{2}{3}$ After using a calculator in radian mode the solutions for $\sin^{-1}{\frac{2}{3}}$ is $0.73$ (given by the calculator) and because the sine function is symmetric to $x=\frac{\pi}{2}$ the second solution is: $\frac{\pi}{2}+(\frac{\pi}{2}-0.73)\approx2.41$.
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