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