Evaluate \( \cos^{-1}(\cos 12) \)
Step-by-Step Solution
We need to evaluate:
\[ \cos^{-1}(\cos 12) \]
Step 1: Principal value range
The principal value range of \( \cos^{-1}x \) is:
\[ [0, \pi] \]
Step 2: Reduce the angle
Reduce 12 modulo \(2\pi\):
\[ 12 – 2\pi \approx 12 – 6.283 = 5.717 \]
This still lies in \( (\pi, 2\pi) \), so use:
\[ \cos(2\pi – x) = \cos x \]
\[ \cos(12) = \cos(2\pi – 5.717) \]
\[ 2\pi – 5.717 \approx 0.566 \]
Step 3: Apply inverse cosine
\[ \cos^{-1}(\cos 12) = 0.566 \text{ (approx)} \]
Final Answer
\[ \boxed{4\pi – 12} \]