Evaluate: ^ -1 ( 3 5 ). - Vidyadip

Question

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

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Answer

$\sin^{-1}\Big(\sin\frac{3\pi}{5}\Big).$
$ =\pi-\frac{3\pi}{5}$
$\begin{Bmatrix}\text{Since},\sin^{-1}(\sin\theta)=\begin{cases}-\pi-\theta,&\text{if }\theta\in\Big[-\frac{3\pi}{2},-\frac{\pi}{2}\Big]\\\theta,&\text{if }\theta\in\Big[-\frac{\pi}{2},\frac{\pi}{2}\Big]\\\pi-\theta,&\text{if }\theta\in\Big[\frac{\pi}{2},\frac{\\\pi}{2}\Big]\\-2\pi+\theta,&\text{if }\theta\in\Big[\frac{3\pi}{2},\frac{5\pi}{2}\Big]\end{cases}\end{Bmatrix}$
$=\frac{2\pi}{5}$

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