Question

The principal value of \( \cot^{-1} \left( -\frac{1}{\sqrt{3}} \right) \) is:

• A. \( -\frac{\pi}{3} \)
• B. \( -\frac{2\pi}{3} \)
• C. \( \frac{\pi}{3} \)
• D. \( \frac{2\pi}{3} \)

Hint:

For \( \cot^{-1}(x) \), the range is \( (0, \pi) \). Make sure to find the correct angle in this range.

Answer 1

The Correct Option is B

We need to find the principal value of \( \cot^{-1} \left( -\frac{1}{\sqrt{3}} \right) \).
The principal value of \( \cot^{-1}(x) \) lies in the range \( (0, \pi) \). For \( \cot \theta = -\frac{1}{\sqrt{3}} \), the corresponding angle \( \theta \) in the principal range is \( \theta = \frac{2\pi}{3} \), since \( \cot \frac{\pi}{3} = \frac{1}{\sqrt{3}} \), and \( \cot \left( \frac{2\pi}{3} \right) = -\frac{1}{\sqrt{3}} \).
Thus, the principal value of \( \cot^{-1} \left( -\frac{1}{\sqrt{3}} \right) \) is \( -\frac{2\pi}{3} \).