Question
Find the value of:
\[ \cos^{-1}(\cos \tfrac{13\pi}{6}) \]
Solution
First, reduce the angle:
\[ \frac{13\pi}{6} = 2\pi + \frac{\pi}{6} \Rightarrow \cos \tfrac{13\pi}{6} = \cos \tfrac{\pi}{6} \]
Now evaluate:
\[ \cos^{-1}(\cos \tfrac{\pi}{6}) \]
The principal value range of \( \cos^{-1}x \) is:
\[ [0, \pi] \]
Since \( \tfrac{\pi}{6} \in [0, \pi] \),
\[ \cos^{-1}(\cos \tfrac{\pi}{6}) = \tfrac{\pi}{6} \]
Final Answer:
\[ \boxed{\tfrac{\pi}{6}} \]
Key Concept
Reduce the angle first and ensure it lies within the principal value range.