Evaluate \( \sin^{-1}(\sin \frac{7\pi}{6}) \)
Step-by-Step Solution
We need to evaluate:
\[ \sin^{-1}\left(\sin \frac{7\pi}{6}\right) \]
Step 1: Find the value of sine
\[ \sin \frac{7\pi}{6} = -\frac{1}{2} \]
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 the angle must lie in this range, we get:
\[ \sin^{-1}\left(-\frac{1}{2}\right) = -\frac{\pi}{6} \]
Final Answer
\[ \boxed{-\frac{\pi}{6}} \]