Evaluate \( \sin^{-1}(\sin \frac{5\pi}{6}) \)
Step-by-Step Solution
We need to evaluate:
\[ \sin^{-1}\left(\sin \frac{5\pi}{6}\right) \]
Step 1: Find the value of sine
\[ \sin \frac{5\pi}{6} = \frac{1}{2} \]
Step 2: Apply inverse sine
\[ \sin^{-1}\left(\frac{1}{2}\right) \]
The principal value range of \( \sin^{-1}x \) is:
\[ \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \]
Since \( \frac{\pi}{6} \) lies in this interval, we get:
\[ \sin^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{6} \]
Final Answer
\[ \boxed{\frac{\pi}{6}} \]