## Precalculus (10th Edition)

$3.48, 5.94$.
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: $\frac{3\pi}{2}+(\frac{3\pi}{2}-5.94)\approx3.48$.