Question
Evaluate:
\[ \sin^{-1}(\sin \tfrac{3\pi}{5}) \]
Solution
The principal value range of \( \sin^{-1}x \) is:
\[ \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \]
Now,
\[ \frac{3\pi}{5} = 108^\circ \]
Since \( \frac{3\pi}{5} \in \left(\frac{\pi}{2}, \pi\right) \), we use identity:
\[ \sin^{-1}(\sin x) = \pi – x \quad \text{for } \frac{\pi}{2} < x < \pi \]
Thus,
\[ \sin^{-1}(\sin \tfrac{3\pi}{5}) = \pi – \frac{3\pi}{5} = \frac{2\pi}{5} \]
Final Answer:
\[ \boxed{\tfrac{2\pi}{5}} \]
Key Concept
Always check whether the angle lies in the principal value range before evaluating.