Evaluate:

Question:

Evaluate: $\sin ^{-1}\left(\sin \frac{3 \pi}{5}\right)$

Solution:

We know that $\sin ^{-1}(\sin x)=x$.

We have

$\sin ^{-1}\left(\sin \frac{3 \pi}{5}\right)=\sin ^{-1}\left\{\sin \left(\pi-\frac{3 \pi}{5}\right)\right\} \quad\left[\because\left(\pi-\frac{3 \pi}{5}\right) \in\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\right]$

$=\sin ^{-1}\left(\sin \frac{2 \pi}{5}\right)$

$=\frac{2 \pi}{5}$

$\therefore \sin ^{-1}\left(\sin \frac{3 \pi}{5}\right)=\frac{2 \pi}{5}$

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