Answer
The exact value of expression is $-\frac{\pi }{6}$.
Work Step by Step
To simplify ${{\sin }^{-1}}\left( \cos \frac{2\pi }{3} \right)$, we apply the property:
$\left( \cos \frac{\pi }{2}+\theta \right)=-\sin \theta $
Therefore,
$\left( \cos \frac{\pi }{2}+\frac{\pi }{6} \right)=-\sin \frac{\pi }{6}$
Then, simplify ${{\sin }^{-1}}\left( -\sin \frac{\pi }{6} \right)$.
Apply the inverse property,
${{\sin }^{-1}}\left( -\sin \frac{\pi }{6} \right)=-\frac{\pi }{6}$
Then, ${{\sin }^{-1}}\left( -\sin \frac{\pi }{6} \right)=-\frac{\pi }{6}$.
Hence, the exact value of the expression is $-\frac{\pi }{6}$.
