Question:
Find the principal value of each of the following :
$\tan ^{-1} 1+\cos ^{-1}\left(-\frac{1}{2}\right)+\sin ^{-1}\left(-\frac{1}{2}\right)$
Solution:
[Formula: $\cos ^{-1}(-x)=\pi-\cos (x)$ and $\sin ^{-1}(-x)=-\sin (x)$ ]
$\tan ^{-1} 1+\left(\pi-\cos ^{-1}\left(\frac{1}{2}\right)\right)+\left(-\sin ^{-1}\left(\frac{1}{2}\right)\right)$
Putting the values for each of the inverse trigonometric terms
$=\frac{\pi}{4}+\left(\pi-\frac{\pi}{3}\right)-\frac{\pi}{6}$
$=\frac{\pi}{12}+\frac{2 \pi}{3}$
$=\frac{9 \pi}{12}$
$=\frac{3 \pi}{4}$