Question
Find the principal value of:
\[ \sin^{-1}\left\{\cos\left(\sin^{-1}\left(\frac{1}{2}\right)\right)\right\} \]
Solution
First, evaluate:
\[ \sin^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{6} \]
So,
\[ \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \]
Now evaluate:
\[ \sin^{-1}\left(\frac{\sqrt{3}}{2}\right) \]
The principal value range of \( \sin^{-1}x \) is:
\[ \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \]
Since \( \frac{\sqrt{3}}{2} \) corresponds to \( \frac{\pi}{3} \) in this range,
\[ \sin^{-1}\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{3} \]
Final Answer:
\[ \boxed{\frac{\pi}{3}} \]
Key Concept
Evaluate step-by-step and always check principal value ranges.