Question

The principal value of \( \sin^{-1} \left( \sin \frac{5\pi}{3} \right) \) is:

• A. \( -\frac{5\pi}{3} \)
• B. \( \frac{5\pi}{3} \)
• C. \( \frac{\pi}{3} \)
• D. \( \frac{4\pi}{3} \)

Hint:

For inverse trigonometric functions, always find the angle within the specified range for the principal value.

Answer 1

The Correct Option is C

Step 1: The principal value of the inverse sine function is in the range \( \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \).
Step 2: \( \sin \frac{5\pi}{3} \) is equivalent to \( -\frac{\pi}{3} \), so the principal value is \( \frac{\pi}{3} \).

Final Answer: \[ \boxed{\frac{\pi}{3}} \]