Question

Find the principal value of ${\cos }^{-1}\left(\frac{\sqrt{3}}{2}\right).$

Answer 1

Let $y={\cos }^{-1}\left(\frac{\sqrt{3}}{2}\right)$
$\Rightarrow \cos y=\frac{\sqrt{3}}{2}$
$\Rightarrow \cos y=\cos \left(\frac{\pi }{6}\right)$
We know that the range of the principal value branch of ${\cos }^{-1}x$ is $[0,\pi ]$ .
$\therefore y=\frac{\pi }{6}$
Hence, the principal value of ${\cos }^{-1}\left(\frac{\sqrt{3}}{2}\right)$ is $\frac{\pi }{6}$