If ^ -1 x = y, then - Vidyadip

Question

If $\sin^{-1} x = y,$ then

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Answer

$\sin^{-1} x = y$
$\Rightarrow \sin y = x$
We know that range of the principle value branch of $\sin^{-1}x$ is $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$
Because, we know that
$-1\le \sin y\le 1$
$\Rightarrow \sin(-\frac {\pi}{2})\le \sin y \le \sin(\frac{\pi}{2})$
Therefore, $-\frac{\pi}{2} \leq y \leq \frac{\pi}{2}$

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