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: Use identity
\[ \sec(-x) = \sec x \]
\[ \sec\left(-\frac{7\pi}{3}\right) = \sec\left(\frac{7\pi}{3}\right) \]
Step 2: 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 3: Apply inverse secant
\[ \sec^{-1}(\sec \frac{\pi}{3}) \]
The principal value range of \( \sec^{-1}x \) is:
\[ [0, \pi] \setminus \left\{\frac{\pi}{2}\right\} \]
Since \( \frac{\pi}{3} \) lies in this range, we get:
\[ \sec^{-1}(\sec -\frac{7\pi}{3}) = \frac{\pi}{3} \]
Final Answer
\[ \boxed{\frac{\pi}{3}} \]