The value of ${\sin ^{ - 1}}\left( {\sin \frac{{3\pi }}{5}} \right)$ is …………….

$- \frac{\pi }{2} \le {\sin ^{ - 1}}x \le \frac{\pi }{2}$

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