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 $\cos ^{-1}\left(\frac{-1}{\sqrt{2}}\right)$ is

A. $\frac{-\pi}{4}$

B. $\frac{\pi}{4}$

C. $\frac{3 \pi}{4}$

D. $\frac{5 \pi}{4}$

 

Solution:

To Find: The Principle value of $\cos ^{-1}\left(\frac{-1}{\sqrt{2}}\right)$

Let the principle value be given by $x$

Now, let $x=\cos ^{-1}\left(\frac{-1}{\sqrt{2}}\right)$

$\Rightarrow \cos x=\frac{-1}{\sqrt{2}}$

$\Rightarrow \cos x=-\cos \left(\frac{\pi}{4}\right)\left(\because \cos \left(\frac{\pi}{4}\right)=\frac{1}{\sqrt{2}}\right)$

$\Rightarrow \cos x=\cos \left(\pi-\frac{\pi}{4}\right)(\because-\cos (\theta)=\cos (\pi-\theta))$

$\Rightarrow x=\frac{3 \pi}{4}$

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