Answer
$\arcsin(-\sqrt{2})$ does not exist
Work Step by Step
$y=\sin^{-1}x =\arcsin x$
Domain: $[-1, 1] $
Range: $[-\displaystyle \frac{\pi}{2}, \displaystyle \frac{\pi}{2}]$
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$y$ is the number from $[-\displaystyle \frac{\pi}{2}, \displaystyle \frac{\pi}{2}]$
such that $\sin y=-\sqrt{2}.$
Such a y can not exist, as
$-1 \leq \sin y \leq 1$
($-\sqrt{2}$ is not in the range of $\sin x,$
$-\sqrt{2}$ is not in the domain of $\sin^{-1}x$)
so
$\arcsin(-\sqrt{2})$ does not exist
