Question

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

Answer 1

Let $y={\cos }^{-1}(-\frac{1}{2})$
$\Rightarrow \cos y=-\frac{1}{2}$
We know that the range of the principal value branch of ${\cos }^{-1}x$ is $[0,\pi ]$.
$\Rightarrow \cos y=\cos \left(\frac{2\pi }{3}\right)$
$\therefore y=\frac{2\pi }{3}$
Hence, the principal value of ${\cos }^{-1}(-\frac{1}{2})$ is $\frac{2\pi }{3}.$