Find the value of tan ^{-1}left(tan frac{7 pi | vedclass

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(B) We know that $	an ^{-1}(	an x) = x$ only if $x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}ight)$,which is the principal value branch of $	an ^{-1

Vedclass · Accepted Answer

$\frac{\pi}{6}$

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(B) We know that $	an ^{-1}(	an x) = x$ only if $x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}ight)$,which is the principal value branch of $	an ^{-1} x$. Here,the given value is $\frac{7 \pi}{6}$,and $\frac{7 \pi}{6} 
otin \left(-\frac{\pi}{2}, \frac{\pi}{2}ight)$. We can simplify the expression using the periodicity of the tangent function: $	an \left(\pi + 	hetaight) = 	an 	heta$. Therefore,$	an \left(\frac{7 \pi}{6}ight) = 	an \left(\pi + \frac{\pi}{6}ight) = 	an \left(\frac{\pi}{6}ight)$. Since $\frac{\pi}{6} \in \left(-\frac{\pi}{2}, \frac{\pi}{2}ight)$,we have $	an ^{-1}\left(	an \frac{7 \pi}{6}ight) = 	an ^{-1}\left(	an \frac{\pi}{6}ight) = \frac{\pi}{6}$.

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

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