Question

Find the principal value of ${\cos }^{-1}(-\frac{1}{2}).$

Answer 1

Let ${\cos }^{-1}(-\frac{1}{2})=y$.

Then, $\cos y=-\frac{1}{2}=\cos \left(\frac{2\pi }{3}\right)$.

We know that the range of the principal value of ${\cos }^{-1}x$ is $[0,\pi ]$ and $\cos \left(\frac{2\pi }{3}\right)=-\frac{1}{2}$

Therefore, the principal value of ${\cos }^{-1}(-\frac{1}{2})$ is $\frac{2\pi }{3}$.