Mark the tick against the correct answer in the following:

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 $x=\sec ^{-1}\left(\frac{-2}{\sqrt{3}}\right)$

$\Rightarrow \sec x=\frac{-2}{\sqrt{3}}$

$\Rightarrow \sec x=-\sec \left(\frac{\pi}{6}\right)\left(\because \sec \left(\frac{\pi}{6}\right)=\frac{2}{\sqrt{3}}\right)$

$\Rightarrow \sec x=\sec \left(\pi-\frac{\pi}{6}\right)(\because-\sec (\theta)=\sec (\pi-\theta))$

$\Rightarrow x=\frac{5 \pi}{6}$

 

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