Question

The principal solution of the equation $\cot x=-\sqrt{3}$ is

• A. $\frac{\pi }{6}$
• B. $\frac{\pi }{3}$
• C. $\frac{5\pi }{6}$
• D. $-\frac{5\pi }{6}$

Answer 1

Answer: (C)

$\frac{5\pi }{6}$

The given equation is
$\cot x=-\sqrt{3}$
$\Rightarrow x={\cot }^{-1}(-\sqrt{3})={\cot }^{-1}\left(\cot (-\frac{\pi }{6})\right)$
We know that, the range of the principal value of ${\cot }^{-1}x$ is $(0,\pi )$
Therefore, $x\in (0,\pi )$.
$\therefore x=\pi -\frac{\pi }{6}=\frac{5\pi }{6}\in (0,\pi )$ and $\cot (\pi -\frac{\pi }{6})=-\cot \frac{\pi }{6}=-\sqrt{3}$
Hence, principal solution of the equation $\cot x=-\sqrt{3}$ is $\frac{5\pi }{6}$.
Therefore, the correct answer from the given alternatives is (c) $\frac{5\pi }{6}$.