Find the values: ^ -1 ( 3 4 ) - Vidyadip

Question

Find the values:
$\tan^{-1}\bigg(\tan\frac{3\pi}{4}\bigg)$

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Answer

For $\tan^{-1}\left(\tan x\right)$ type of problem we have to always check whether the angle is in the principle range or not. This angle must be in the principle range $\bigg[-\frac{\pi}{2},\frac{\pi}{2}\bigg].$
$=\tan^{-1}\bigg(\tan\frac{4\pi-\pi}{4}\bigg)$
$=\tan^{-1}\bigg[\tan\bigg(\pi-\frac{\pi}{4}\bigg)\bigg]=\tan^{-1}\bigg[-\tan\frac{\pi}{4}\bigg]$
$=\tan^{-1}\tan\bigg(-\frac{\pi}{4}\bigg)=-\frac{\pi}{4}$

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