Answer
$\sin^{-1}(-\sqrt{2})$ is not defined.
Work Step by Step
Inverse Sine Function
$y=\sin^{-1}x$ or $y=$ arcsin $x$ means that
$x=\sin y$, for $-\displaystyle \frac{\pi}{2} \leq y \leq \frac{\pi}{2}$.
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Since $-\sqrt{2} < 1$, there is no y such that
$ \sin y=\sqrt{-2}$
$y=\sin^{-1}(-\sqrt{2})$ is not defined.
$\sin^{-1}$or arcsin is the inverse of $\sin,$ so its domain must be the range of sine.
The range of sine is $[-1,1]$ and $-\sqrt{2}\not\in [-1,1] )$
