Question:
Mark the tick against the correct answer in the following:
The principal value of $\sec ^{-1}\left(\frac{-2}{\sqrt{3}}\right)$ is
A. $\frac{\pi}{6}$
B. $\frac{-\pi}{6}$
C. $\frac{5 \pi}{6}$
D. $\frac{7 \pi}{6}$
Solution:
To Find: The Principle value of $\sec ^{-1}\left(\frac{-2}{\sqrt{3}}\right)$
Let the principle value be given by $x$
Now, let $x=\sec ^{-1}\left(\frac{-2}{\sqrt{3}}\right)$
$\Rightarrow \sec x=\frac{-2}{\sqrt{3}}$
$\Rightarrow \sec x=-\sec \left(\frac{\pi}{6}\right)\left(\because \sec \left(\frac{\pi}{6}\right)=\frac{2}{\sqrt{3}}\right)$
$\Rightarrow \sec x=\sec \left(\pi-\frac{\pi}{6}\right)(\because-\sec (\theta)=\sec (\pi-\theta))$
$\Rightarrow x=\frac{5 \pi}{6}$