Question

20 Find the value of $\tan ^{-1}\left(\tan \frac{3 \pi}{4}\right)$.

Answer 1

SOLUTION :    $\because \tan ^{-1}(\tan x)=x \quad \text { if } x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \\$

$\text { As } \quad \frac{3 \pi}{4} \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \quad \therefore \quad \tan ^{-1}\left(\tan \frac{3 \pi}{4}\right) \neq \frac{3 \pi}{4} \\$

$\therefore \quad \frac{3 \pi}{4} \in\left(\frac{\pi}{2}, \frac{3 \pi}{2}\right)$

graph of $y=\tan ^{-1}(\tan x)$ is as :

$\because \quad$ from the graph we can see that if $\frac{\pi}{2}<x<\frac{3 \pi}{2}$,

then $\tan ^{-1}(\tan x)=x-\pi$

$\therefore \quad \tan ^{-1}\left(\tan \frac{3 \pi}{4}\right)=\frac{3 \pi}{4}-\pi=-\frac{\pi}{4}$