Question

Principle value of $\cot^{-1}(-1)$ is

• A. $\frac{2\pi}{3}$
• B. $-\frac{2\pi}{3}$
• C. $\pi$
• D. $\frac{3\pi}{4}$

Hint:

Always remember the range of inverse functions. For $\cos^{-1}x$, $\sec^{-1}x$, and $\cot^{-1}x$, the range is $[0, \pi]$ or $(0, \pi)$. If the input is negative, the answer will always be an obtuse angle (Quadrant II).

Answer 1

The Correct Option is D

The principal value branch of $\cot^{-1} x$ is $(0, \pi)$. 1. Setting the Equation: Let $y = \cot^{-1}(-1)$. This implies: $$\cot y = -1$$

2. Finding the Angle: We know that $\cot(\frac{\pi}{4}) = 1$. Since the value is negative, the angle must lie in the second quadrant (as per the range $(0, \pi)$). $$\cot y = -\cot\left(\frac{\pi}{4}\right)$$ Using the identity $\cot(\pi - \theta) = -\cot \theta$: $$\cot y = \cot\left(\pi - \frac{\pi}{4}\right)$$ $$\cot y = \cot\left(\frac{3\pi}{4}\right)$$

3. Conclusion: Since $\frac{3\pi}{4}$ lies within the interval $(0, \pi)$, the principal value is $\frac{3\pi}{4}$.