The principal value of ^ -1 (-3) is - Vidyadip

MCQ

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

A
$-\frac{\pi}{6}$
B
$\frac{\pi}{6}$
C
$\frac{2 \pi}{3}$
D
$\frac{5 \pi}{6}$

✓

Answer

We know that $\cot ^{-1}(x) \in(0, \pi)$
\[\begin{array}{ll}
\cot ^{-1}(-\sqrt{3})=\cot ^{-1}\left(-\cot \frac{\pi}{6}\right) & \\
=\cot ^{-1}\left[\cot \left(\pi-\frac{\pi}{6}\right)\right] & {[\because \cot (\pi-\theta)=-\cot \theta]} \\
=\cot ^{-1}\left[\cot \left(\frac{5 \pi}{6}\right)\right]=\frac{5 \pi}{6} & {\left[\because \cot ^{-1}[\cot \theta]=\theta\right]}
\end{array}
\]Thus, the principal value of $\cot ^{-1}(-\sqrt{3})$ is $\frac{5 \pi}{6}$.

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