Question
Find the value of:
\[ \cos^{-1}(\cos 6) \]
Solution
We know the principal value range of \( \cos^{-1}x \) is:
\[ [0, \pi] \]
Now, note that:
\[ 2\pi \approx 6.283 \Rightarrow 6 = 2\pi – 0.283 \]
So,
\[ \cos 6 = \cos(2\pi – 0.283) = \cos(0.283) \]
Thus,
\[ \cos^{-1}(\cos 6) = \cos^{-1}(\cos 0.283) \]
Since \( 0.283 \in [0, \pi] \),
\[ = 0.283 \]
Also,
\[ 0.283 = 2\pi – 6 \]
Final Answer:
\[ \boxed{2\pi – 6} \]
Key Concept
Reduce the angle into the principal range \([0, \pi]\) before applying inverse cosine.