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