Question:
Mark the tick against the correct answer in the following:
The principal value of $\cot ^{-1}(-\sqrt{3})$ is
A. $\frac{2 \pi}{6}$
B. $\frac{\pi}{6}$
C. $\frac{7 \pi}{6}$
D. $\frac{5 \pi}{6}$
Solution:
To Find: The Principle value of $\cot ^{-1}(-\sqrt{3})$
Let the principle value be given by $x$
Now, let $x=\cot ^{-1}(-\sqrt{3})$
$\Rightarrow \cot x=-\sqrt{3}$
$\Rightarrow \cot x=-\cot \left(\frac{\pi}{6}\right)\left(\because \cot \left(\frac{\pi}{6}\right)=\sqrt{3}\right)$
$\Rightarrow \cot x=\cot \left(\pi-\frac{\pi}{6}\right)(\because-\cot (\theta)=\cot (\pi-\theta))$
$\Rightarrow x=\frac{5 \pi}{6}$