Question

Find the principal values of the following questions:

$co{s}^{-1}\left(\frac{\sqrt{3}}{2}\right)$

Answer 1

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