Evaluate \( \cos^{-1}(\cos 4) \)
Step-by-Step Solution
We need to evaluate:
\[ \cos^{-1}(\cos 4) \]
Step 1: Principal value range
The principal value range of \( \cos^{-1}x \) is:
\[ [0, \pi] \]
Step 2: Locate 4
Since \( 4 \in (\pi, 2\pi) \), we use identity:
\[ \cos(2\pi – x) = \cos x \]
\[ \cos(4) = \cos(2\pi – 4) \]
Step 3: Apply inverse cosine
\[ \cos^{-1}(\cos 4) = \cos^{-1}(\cos(2\pi – 4)) = 2\pi – 4 \]
Now \( 2\pi – 4 \in [0, \pi] \), so it is valid.
Final Answer
\[ \boxed{2\pi – 4} \]