Answer
The exact value of the provided expression is $\frac{2\pi }{3}$.
Work Step by Step
Let us assume $\theta ={{\cos }^{-1}}\left( -\frac{1}{2} \right)$. Then, we have
$\cos \theta =\left( -\frac{1}{2} \right)$
We know that for the cosine function, the interval for the angle is $\left[ 0,\pi \right]$.
Therefore, the only angle that satisfies $\cos \theta =\left( -\frac{1}{2} \right)$ is $\left( \frac{2\pi }{3} \right)$.
Hence, $\theta =\frac{2\pi }{3}$ and the exact value of the given expression is
${{\cos }^{-1}}\left( -\frac{1}{2} \right)=\frac{2\pi }{3}$
