Evaluate \( \sec^{-1}(\sec \frac{2\pi}{3}) \)
Step-by-Step Solution
We need to evaluate:
\[ \sec^{-1}\left(\sec \frac{2\pi}{3}\right) \]
Step 1: Convert to cosine
\[ \sec x = \frac{1}{\cos x} \]
\[ \cos \frac{2\pi}{3} = -\frac{1}{2} \Rightarrow \sec \frac{2\pi}{3} = -2 \]
Step 2: Apply inverse secant
\[ \sec^{-1}(-2) \]
Step 3: Use principal value range
The principal value range of \( \sec^{-1}x \) is:
\[ [0, \pi] \setminus \left\{\frac{\pi}{2}\right\} \]
Step 4: Find the correct angle
\[ \sec \theta = -2 \Rightarrow \cos \theta = -\frac{1}{2} \]
In the interval \( [0, \pi] \), this occurs at:
\[ \theta = \frac{2\pi}{3} \]
Final Answer
\[ \boxed{\frac{2\pi}{3}} \]