Find the value of \({\sin ^{ - 1}}\left( {\sin \frac{{4\pi }}{5}} \right)\)

Concept:

\({\sin ^{ - 1}}\ (sin x) = x\) if \(x \in \left[ { - \frac{\pi }{2},\frac{\pi }{2}} \right]\) but if \(x{ \notin }\left[ { - \frac{\pi }{2},\frac{\pi }{2}} \right]\), then use \(\sin x = \sin \left( {\pi - x} \right)\) to bring the value of x inside the principle branch.

Solution:

\({\sin ^{ - 1}}\left( {\sin \frac{{4\pi }}{5}} \right)\) but \(\frac{{4\pi }}{5}\notin{ }\left[ { - \frac{\pi }{2},\frac{\pi }{2}} \right]\)

So, use the relation,

\(\sin \frac{{4\pi }}{5} = \sin \left( {\pi - \frac{{4\pi }}{5}} \right)\)

\(= \sin \left( {\frac{\pi }{5}} \right)\)

So,

\({\sin ^{ - 1}}\left( {\sin \frac{{4\pi }}{5}} \right) = {\sin ^{ - 1}}\left( {\sin \frac{\pi }{5}} \right)\)

\( = \frac{\pi }{5}\)