Precalculus (6th Edition)

$-90^{o}$
Solve for radians, then convert to degrees. $y=\csc^{-1}x$ Domain: $(-\infty, -1]\cup[1, \infty)$ Range: $[-\displaystyle \frac{\pi}{2},0)\cup(0,\frac{\pi}{2}]$ Quadrants (unit circle): I and IV --------- $y$ is the number from $[-\displaystyle \frac{\pi}{2},0)\cup(0,\frac{\pi}{2}]$ such that $\csc y=-1\quad (\sin y=-1)$ $\displaystyle \sin-\frac{\pi}{2}=\frac{1}{2}$, , that is, $\csc (-\displaystyle \frac{\pi}{2})=-1$ and $-\displaystyle \frac{\pi}{2}\in [-\displaystyle \frac{\pi}{2},0)\cup(0,\frac{\pi}{2}]$. So, $y=-\displaystyle \frac{\pi}{2}$ To convert radians to degrees, multiply y with $\displaystyle \frac{180^{o}}{\pi}$ $\displaystyle \theta=-\frac{\pi}{2}\cdot\frac{180^{o}}{\pi}=-90^{o}$