Question:
Find the principal value of each of the following :
$\tan ^{-1}\left(\tan \frac{7 \pi}{6}\right)$
Solution:
$\tan ^{-1}\left(\tan \frac{7 \pi}{6}\right)=\tan ^{-1}\left(\tan \left(\pi+\frac{\pi}{6}\right)\right)$
[ Formula: $\tan (\pi+x)=\tan x$, as tan is positive in the third quadrant.]
$=\tan ^{-1}\left(\tan \frac{\pi}{6}\right)\left[\right.$ Formula: $\left.\tan ^{-1}(\tan x)=x\right]$
$=\frac{\pi}{6}$
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