Find the value ^ -1 ( 2 3 ) - Vidyadip

Question

Find the value $\tan ^{-1}\left(\tan \frac{2 \pi}{3}\right)$

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Answer

We have,
$\tan ^{-1}\left(\tan \frac{2 \pi}{3}\right)=\tan ^{-1} \tan \left(\pi-\frac{\pi}{3}\right)$
$ =\tan ^{-1}\left(-\tan \frac{\pi}{3}\right)\left[\because \tan ^{-1}(-x)=-\tan ^{-1} x\right]$
$=\tan ^{-1} \tan \left(\frac{-\pi}{3}\right)=-\frac{\pi}{3}\left[\because \tan ^{-1}(\tan x)=x, x \in\left(\frac{-\pi}{2}, \frac{\pi}{2}\right)\right] $
Note: Remember that, $\tan ^{-1}\left(\tan \frac{2 \pi}{3}\right) \neq \frac{2 \pi}{3}$
Since, $\tan ^{-1}(\tan x )= x$, if $x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ and $\frac{2 \pi}{3} \notin\left(\frac{-\pi}{2}, \frac{\pi}{2}\right)$

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