Answer
$-\displaystyle \frac{\pi}{2}$
Work Step by Step
$y=\sin^{-1}x$
Domain: $[-1, 1] $Range: $[-\displaystyle \frac{\pi}{2}, \displaystyle \frac{\pi}{2}]$
$y$ is the number from $[-\displaystyle \frac{\pi}{2}, \displaystyle \frac{\pi}{2}]$ such that $\sin y=-1$
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$\displaystyle \sin(-\frac{\pi}{2})=-1$, and $-\displaystyle \frac{\pi}{2}\in[-\frac{\pi}{2}, \displaystyle \frac{\pi}{2}]$ so
$\displaystyle \sin^{-1}(-1)=-\frac{\pi}{2}$
