## Precalculus (6th Edition) Blitzer

The exact value of expression is $-\frac{\pi }{6}$.
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 $-\frac{\pi }{6}$.