Question

Find the principal values of the following questions:

$se{c}^{-1}\left(\frac{2}{\sqrt{3}}\right)$

Answer 1

Let $se{c}^{-1}\left(\frac{2}{\sqrt{3}}\right)=\theta \Rightarrow sec\theta =\frac{2}{\sqrt{3}}$
We know that the range of principal value of $se{c}^{-1}\theta is[0,\pi ]-\left\{\frac{\pi }{2}\right\}$

$\therefore sec\theta =\frac{2}{\sqrt{3}}=sec\frac{\pi }{6}\Rightarrow \theta =\frac{\pi }{6}$, where $\theta ϵ[0,\pi ]-\left\{\frac{\pi }{2}\right\}$

$\Rightarrow se{c}^{-1}\left(\frac{2}{\sqrt{3}}\right)=\frac{\pi }{6}$
Hence, the principal value of $se{c}^{-1}\left(\frac{2}{\sqrt{3}}\right)$ is $\frac{\pi }{6}$