Question:
Find the principal value of:
\[ \sin^{-1}\left(\cos \frac{2\pi}{3}\right) \]
Solution:
Step 1: Evaluate cosine value
\[ \cos \frac{2\pi}{3} = -\frac{1}{2} \]
Step 2: Substitute
\[ \sin^{-1}\left(-\frac{1}{2}\right) \]
Step 3: Use standard value
\[ \sin\left(-\frac{\pi}{6}\right) = -\frac{1}{2} \]
Step 4: Check principal range
\[ -\frac{\pi}{6} \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \]
—Final Answer:
\[ \boxed{-\frac{\pi}{6}} \]