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: 54


$3.48, 5.94$.

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. $\left[− \frac{\pi}{2} , \frac{\pi}{2} \right]$. We know that $\csc^{-1}(x)=\sin^{-1}{\frac{1}{x}}$. After using a calculator in radian mode the solutions for $\csc^{-1}{-3}=\sin^{-1}{-\frac{1}{3}}$ is $-0.34$ (given by the calculator) and because the csc function has a period of $2\pi$, the first solution is $-0.34+2\pi=5.94.$ Because the csc function is symmetric to $x=\frac{3\pi}{2}$ the second solution is: $\frac{3\pi}{2}+(\frac{3\pi}{2}-5.94)\approx3.48$.
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