Question

The principal value of \( \sin^{-1} \left( -\frac{1}{2} \right) \) is

• A. \( \frac{\pi}{3} \)
• B. \( \frac{\pi}{6} \)
• C. \( -\frac{\pi}{3} \)
• D. \( -\frac{\pi}{6} \)

Hint:

For inverse trigonometric functions, remember that the principal value is the value of the angle within the specified range.

Answer 1

The Correct Option is D

Step 1: Use the inverse sine function.
For \( \sin^{-1} \left( -\frac{1}{2} \right) \), the principal value lies in the interval \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \). We know that \( \sin \left( -\frac{\pi}{6} \right) = -\frac{1}{2} \). Therefore, the principal value is \( -\frac{\pi}{6} \).
Step 2: Conclusion.
The correct answer is (D) \( -\frac{\pi}{6} \).