Question

Find the principal value of ${\tan }^{-1}(-\sqrt{3})$

Answer 1

Let $y={\tan }^{-1}(-\sqrt{3})$
$\Rightarrow \tan y=-\sqrt{3}$
We know that the range of the principal value branch of ${\tan }^{-1}$ is $(-\frac{\pi }{2},\frac{\pi }{2})$.
$\Rightarrow \tan y=\tan \left(\frac{-\pi }{3}\right)$
$\therefore y=-\frac{\pi }{3}$
Hence, the principal value of ${\tan }^{-1}(-\sqrt{3})$ is $-\frac{\pi }{3}$