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 check the principal range and adjust the angle accordingly.

Answer 1

The Correct Option is C

Step 1: First, calculate \( \sin \frac{5\pi}{3} \). Since \( \frac{5\pi}{3} \) is in the fourth quadrant, the sine of this angle is negative.
We know that: \[ \frac{5\pi}{3} = 2\pi - \frac{\pi}{3}. \] Thus, \[ \sin \frac{5\pi}{3} = \sin \left( 2\pi - \frac{\pi}{3} \right) = -\sin \frac{\pi}{3} = -\frac{\sqrt{3}}{2}. \] Step 2: Now, find the principal value of \( \sin^{-1} \left( -\frac{\sqrt{3}}{2} \right) \). The principal value of the inverse sine function lies between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \). We know that \( \sin \left( -\frac{\pi}{3} \right) = -\frac{\sqrt{3}}{2} \), so: \[ \sin^{-1} \left( -\frac{\sqrt{3}}{2} \right) = -\frac{\pi}{3}. \] Thus, the answer is \( -\frac{\pi}{3} \).