Question

Find the value of ${\sin }^{-1}\left(\sin \frac{3\pi }{5}\right).$

Answer 1

Given :
${\sin }^{-1}\left(\sin \frac{3\pi }{5}\right)$
We know that ${\sin }^{-1}\left(\sin x\right)=x,x\in [-\frac{\pi }{2},\frac{\pi }{2}]$
Since $\frac{3\pi }{5}\notin [-\frac{\pi }{2},\frac{\pi }{2}]$, which is the range of principal value branch of ${\sin }^{-1}$
However, $\sin \left(\frac{3\pi }{5}\right)=\sin (\pi -\frac{3\pi }{5})=\sin \frac{2\pi }{5}$
and $\frac{2\pi }{5}\in [-\frac{\pi }{2},\frac{\pi }{2}]$
$\therefore {\sin }^{-1}\left(\sin \frac{3\pi }{5}\right)={\sin }^{-1}\left(\sin \frac{2\pi }{5}\right)=\frac{2\pi }{5}$