Question
Find the value of:
\[ \cos^{-1}(\cos \tfrac{14\pi}{3}) \]
Solution
First, reduce the angle using periodicity:
\[ \frac{14\pi}{3} = 4\pi + \frac{2\pi}{3} \Rightarrow \cos \tfrac{14\pi}{3} = \cos \tfrac{2\pi}{3} \]
Now evaluate:
\[ \cos^{-1}(\cos \tfrac{2\pi}{3}) \]
The principal value range of \( \cos^{-1}x \) is:
\[ [0, \pi] \]
Since \( \tfrac{2\pi}{3} \in [0, \pi] \),
\[ \cos^{-1}(\cos \tfrac{2\pi}{3}) = \tfrac{2\pi}{3} \]
Final Answer:
\[ \boxed{\tfrac{2\pi}{3}} \]
Key Concept
Reduce the angle first and ensure it lies within the principal value range.