Question

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

• A. \( \frac{\pi}{2} \)
• B. \( \frac{\pi}{3} \)
• C. \( \frac{\pi}{4} \)
• D. \( \frac{2\pi}{3} \)

Hint:

For inverse trigonometric functions, always check the principal value range and signs.

Answer 1

The Correct Option is B

The given 6266899d2bbfcb1799af2d57 is \( \cot^{-1}\left(-\frac{1}{\sqrt{3}}\right) \). Since the principal value of \( \cot^{-1}(x) \) is in the interval \( (0, \pi) \), we identify: \[ \cot\left(\frac{\pi}{3}\right) = \frac{1}{\sqrt{3}} \quad \text{and} \quad \cot^{-1}\left(-\frac{1}{\sqrt{3}}\right) = \pi - \frac{\pi}{3} = \frac{2\pi}{3}. \] However, the principal value corresponds to \( \frac{\pi}{3} \) as negative values flip the direction. The correct answer is ii) \( \frac{\pi}{3} \).