Find the principal value of ^ -1 (1 2 ) - Vidyadip

Question

Find the principal value of $\sin ^{-1}\left(\frac{1}{\sqrt{2}}\right)$

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Answer

Let $\sin ^{-1}\left(\frac{1}{\sqrt{2}}\right)$
$\therefore \sin ^{-1} x=\left(\frac{1}{\sqrt{2}}\right)$
$\therefore \sin x=\sin \left(\frac{\pi}{4}\right)$
The principal value branch of
$\sin ^{-1} x$ is $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$
$\sin ^{-1}(\sin \theta)=\theta$
Hence, the required principal value of $x$ is $\frac{\pi}{4}$

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