Question

Find the principal value of ${\cot }^{-1}\left(\frac{-1}{\sqrt{3}}\right).$

Answer 1

Let $y={\cot }^{-1}\left(\frac{-1}{\sqrt{3}}\right)$
$\Rightarrow \cot y=\frac{-1}{\sqrt{3}}$
$\Rightarrow \cot y=\cot (\pi -\frac{\pi }{3})$
We know that the range of the principal value branch of ${\cot }^{-1}x$ is $(0,\pi )$
$\Rightarrow y=\pi -\frac{\pi }{3}$
$\Rightarrow y=\frac{2\pi }{3}$
Hence, the principal value of ${\cot }^{-1}\left(\frac{-1}{\sqrt{3}}\right)$ is $\frac{2\pi }{3}$