Question:
Find the principal value of each of the following :
$\cos ^{-1}\left(\cos \frac{7 \pi}{6}\right)$
Solution:
$\cos ^{-1}\left(\cos \frac{7 \pi}{6}\right)=\cos ^{-1}\left(\cos \left(2 \pi-\frac{5 \pi}{6}\right)\right)$
[Formula: $\cos (2 \pi-x)=\cos (x)$, as $\cos$ has a positive vaule in the fourth quadrant. ]
$=\cos ^{-1}\left(\cos \frac{5 \pi}{6}\right)\left[\right.$ Formula: $\cos ^{-1}(\cos x)=x$
$=\frac{5 \pi}{6}$