Find the principal value of each of the following :

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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