Find the value of ^ - 1 ( 7 6 ) - Vidyadip

Question

Find the value of ${\tan ^{ - 1}}\left( {\tan \frac{{7\pi }}{6}} \right)$

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Answer

${\tan ^{ - 1}}\left( {\tan \frac{{7\pi }}{6}} \right)$ $\ne\frac{7π}{6}$ as principal value branch of $\tan^{-1}$ is $(\frac{-π}{2},\frac{π}{2})$

= So, $tan^{-1}(\frac{7π}{6})$

= ${\tan ^{ - 1}}\left[ {\tan \left( {\pi + \frac{\pi }{6}} \right)} \right]$

$$= ${\tan ^{ - 1}}\left( {\tan \frac{{\pi }}{6}} \right)$

=$\frac{\pi}{6}$ $$ $$

$\therefore$ $tan^{-1}(tan\frac{7π}{6})$ = $\frac{π}{6}$.

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