Evaluate \( \cos^{-1}(\cos \frac{13\pi}{6}) \)
Step-by-Step Solution
We need to evaluate:
\[ \cos^{-1}\left(\cos \frac{13\pi}{6}\right) \]
Step 1: Reduce the angle
\[ \frac{13\pi}{6} = 2\pi + \frac{\pi}{6} \]
\[ \cos\left(\frac{13\pi}{6}\right) = \cos\left(\frac{\pi}{6}\right) \]
Step 2: Apply inverse cosine
\[ \cos^{-1}\left(\cos \frac{\pi}{6}\right) \]
The principal value range of \( \cos^{-1}x \) is:
\[ [0, \pi] \]
Since \( \frac{\pi}{6} \) lies in this range, we get:
\[ \cos^{-1}\left(\cos \frac{13\pi}{6}\right) = \frac{\pi}{6} \]
Final Answer
\[ \boxed{\frac{\pi}{6}} \]