Evaluate \( \sec^{-1}(\sec \frac{7\pi}{3}) \)
Step-by-Step Solution
We need to evaluate:
\[ \sec^{-1}\left(\sec \frac{7\pi}{3}\right) \]
Step 1: Reduce the angle
\[ \frac{7\pi}{3} = 2\pi + \frac{\pi}{3} \]
\[ \sec\left(\frac{7\pi}{3}\right) = \sec\left(\frac{\pi}{3}\right) \]
Step 2: Convert to cosine
\[ \sec \frac{\pi}{3} = 2 \Rightarrow \cos \theta = \frac{1}{2} \]
Step 3: Apply inverse secant
The principal value range of \( \sec^{-1}x \) is:
\[ [0, \pi] \setminus \left\{\frac{\pi}{2}\right\} \]
Now find angle in this range:
\[ \cos \theta = \frac{1}{2} \Rightarrow \theta = \frac{\pi}{3} \]
Final Answer
\[ \boxed{\frac{\pi}{3}} \]