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 $x=\cos ^{-1}\left(\frac{-1}{\sqrt{2}}\right)$
$\Rightarrow \cos x=\frac{-1}{\sqrt{2}}$
$\Rightarrow \cos x=-\cos \left(\frac{\pi}{4}\right)\left(\because \cos \left(\frac{\pi}{4}\right)=\frac{1}{\sqrt{2}}\right)$
$\Rightarrow \cos x=\cos \left(\pi-\frac{\pi}{4}\right)(\because-\cos (\theta)=\cos (\pi-\theta))$
$\Rightarrow x=\frac{3 \pi}{4}$