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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