Question

The principal value of ${\tan }^{-1}(-\sqrt{3})$ is

• A. $\frac{2\pi }{3}$
• B. $\frac{\pi }{3}$
• C. $-\frac{\pi }{3}$
• D. $-\frac{\pi }{6}$

Answer 1

The correct option is C $-\frac{\pi }{3}$

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