^ -1 (-1 2 ) - Vidyadip

Question

$\sin ^{-1}\left(\frac{-1}{2}\right)$

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Answer

Let $\sin ^{-1}\left(\frac{-1}{2}\right)=y$
$\Rightarrow \sin y=-\frac{1}{2}$
$\Rightarrow \sin y=-\sin \frac{\pi}{6}$
$\Rightarrow \sin y=\sin \left(-\frac{\pi}{6}\right)$
Since, the principal value branch of $\sin ^{-1}$ is $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$
Therefore, principal value of $\sin ^{-1}\left(\frac{-1}{2}\right)$ is $-\frac{\pi}{6}$.

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