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 ${\sin ^{ - 1}}\left( {\frac{1}{{\sqrt 2 }}} \right) = \theta$
$\implies\sin \theta = \frac{1}{{\sqrt 2 }}$
We know that $\theta \in \left[ {\frac{{ - \pi }}{2},\frac{\pi }{2}} \right]$
$\implies\sin \theta = \sin \frac{\pi }{4}$
$\implies\theta = \frac{\pi }{4}$
Therefore, principal value of ${\sin ^{ - 1}}\left( {\frac{1}{{\sqrt 2 }}} \right)$ is $\frac{\pi }{4}$

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