Answer
The exact value of the provided expression is $-\frac{\pi }{3}$.
Work Step by Step
Let us assume $\theta ={{\sin }^{-1}}\left( -\frac{\sqrt{3}}{2} \right)$. Then, we have
$\sin \theta =\left( -\frac{\sqrt{3}}{2} \right)$
We know that for the sine function, the interval for the angle is $\left[ \frac{-\pi }{2},\frac{\pi }{2} \right]$.
So, the only angle that satisfies $\sin \theta =\left( -\frac{\sqrt{3}}{2} \right)$ is $\left( -\frac{\pi }{3} \right)$.
Hence, $\theta =-\frac{\pi }{3}$ and the exact value of the given expression is:
${{\sin }^{-1}}\left( -\frac{\sqrt{3}}{2} \right)=-\frac{\pi }{3}$
