Question

\( \sin^{-1} \left(\sin \frac{3\pi}{5}\right) =\) _____

• A. \( \frac{\pi}{5} \)
• B. \( \frac{3\pi}{5} \)
• C. \( \frac{2\pi}{5} \)
• D. \( \frac{4\pi}{5} \)

Hint:

Convert angles into principal range before applying inverse trig functions.

Answer 1

The Correct Option is C

Concept: \[ \sin^{-1}x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \]
Step 1: Given angle. \[ \frac{3\pi}{5} = \pi - \frac{2\pi}{5} \]
Step 2: \[ \sin \frac{3\pi}{5} = \sin \left(\pi - \frac{2\pi}{5}\right) = \sin \frac{2\pi}{5} \]
Step 3: \[ \sin^{-1} \left(\sin \frac{3\pi}{5}\right) = \frac{2\pi}{5} \]