Question

The principal value of \( \tan^{-1}(\sqrt{3}) \) is:

• A. \( -\frac{\pi}{6} \)
• B. \( \frac{\pi}{3} \)
• C. \( -\frac{\pi}{3} \)
• D. None of these

Hint:

The principal value of \( \tan^{-1}(x) \) is the angle in the range \( \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \) for which \( \tan(\theta) = x \).

Answer 1

The Correct Option is B

Step 1: Understanding the inverse tangent function. 
The principal value of \( \tan^{-1}(x) \) is the angle \( \theta \) in the range \( \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \) such that \( \tan(\theta) = x \). 
Step 2: Applying the inverse tangent to \( \sqrt{3} \). 
We know that: \[ \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \] Step 3: Conclusion. 
Thus, the principal value of \( \tan^{-1}(\sqrt{3}) \) is \( \frac{\pi}{3} \), corresponding to option (b).