Question:
Mark the tick against the correct answer in the following:
The principal value of $\cos ^{-1}\left(\frac{-1}{2}\right)$ is
A. $\frac{-\pi}{3}$
B. $\frac{2 \pi}{3}$
C. $\frac{4 \pi}{3}$
D. $\frac{\pi}{3}$
Solution:
To Find: The Principle value of $\cos ^{-1}\left(\frac{-1}{2}\right)$
Let the principle value be given by $\mathrm{x}$
Now, let $x=\cos ^{-1}\left(\frac{-1}{2}\right)$
$\Rightarrow \cos x=\frac{-1}{2}$
$\Rightarrow \cos x=-\cos \left(\frac{\pi}{3}\right)\left(\because \cos \left(\frac{\pi}{3}\right)=\frac{1}{2}\right)$
$\Rightarrow \cos x=\cos \left(\pi-\frac{\pi}{3}\right)(\because-\cos (\theta)=\cos (\pi-\theta))$
$\Rightarrow x=\frac{2 \pi}{3}$