The principal value of ^ -1 ( 3 5 ) is

MCQ

The principal value of $\tan ^{-1}\left(\tan \frac{3 \pi}{5}\right)$ is

A
$\frac{2 \pi}{5}$
B
$\frac{-2 \pi}{5}$
C
$\frac{3 \pi}{5}$
D
$\frac{-3 \pi}{5}$

✓

Answer

We have, $\tan ^{-1}\left(\tan \frac{3 \pi}{5}\right)$
We know that the range of $\tan ^{-1} x$ is $\left(\frac{-\pi}{2}, \frac{\pi}{2}\right)$
\[\begin{array}{l}
\therefore \tan ^{-1}\left(\tan \frac{3 \pi}{5}\right)=\tan ^{-1}\left(\tan \left(\pi-\frac{2 \pi}{5}\right)\right) \\
=\tan ^{-1}\left[-\tan \left(\frac{2 \pi}{5}\right)\right] \quad[\because \tan (\pi-\theta)=\tan \theta] \\
=-\tan ^{-1}\left[\tan \left(\frac{2 \pi}{5}\right)\right]=-\frac{2 \pi}{5} \quad\left[\because \tan ^{-1}(\tan \theta)=\theta\right]
\end{array}\]

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