Question

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

Hint:

For \( \cot^{-1}(x) \), principal value is in \( (0, \pi) \); use \( \cot (\pi - \theta) = -\cot \theta \) for negative arguments.

Answer 1

The principal value of \( \cot^{-1}(x) \) lies in \( (0, \pi) \).
Let \( \theta = \cot^{-1} \left( -\frac{1}{\sqrt{3}} \right) \), so \( \cot \theta = -\frac{1}{\sqrt{3}} \).
Since \( \cot \theta \) is negative, \( \theta \) is in the second quadrant (\( \frac{\pi}{2} < \theta < \pi \)).
Consider: \( \cot \theta = -\cot \left( \frac{\pi}{3} \right) = -\frac{1}{\sqrt{3}} \), since \( \cot \frac{\pi}{3} = \frac{1}{\sqrt{3}} \).
Thus, \( \theta = \pi - \frac{\pi}{3} = \frac{2\pi}{3} \).
Check: \( \cot \frac{2\pi}{3} = \cot (180^\circ - 60^\circ) = -\cot 60^\circ = -\frac{1}{\sqrt{3}} \), and \( \frac{2\pi}{3} \in (0, \pi) \).
Answer: \( \frac{2\pi}{3} \).