Question:
Mark the tick against the correct answer in the following:
The principal value of $\tan ^{-1}(-\sqrt{3})$ is
A. $\frac{2 \pi}{3}$
B. $\frac{4 \pi}{3}$
C. $\frac{-\pi}{3}$
D. none of these
Solution:
To Find: The Principle value of $\tan ^{-1}(-\sqrt{3})$
Let the principle value be given by $x$
Now, let $x=\tan ^{-1}(-\sqrt{3})$
$\Rightarrow \tan x=-\sqrt{3}$
$\Rightarrow \tan x=-\tan \left(\frac{\pi}{3}\right)\left(\because \tan \left(\frac{\pi}{3}\right)=-\sqrt{3}\right)$
$\Rightarrow \tan x=\tan \left(-\frac{\pi}{3}\right)(\because-\tan (\theta)=\tan (-\theta))$
$\Rightarrow x=-\frac{\pi}{3}$
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