Evaluate \( \sec^{-1}(\sec \frac{13\pi}{4}) \)
Step-by-Step Solution
We need to evaluate:
\[ \sec^{-1}\left(\sec \frac{13\pi}{4}\right) \]
Step 1: Reduce the angle
\[ \frac{13\pi}{4} = 2\pi + \frac{5\pi}{4} \]
\[ \sec\left(\frac{13\pi}{4}\right) = \sec\left(\frac{5\pi}{4}\right) \]
Step 2: Convert to cosine
\[ \cos \frac{5\pi}{4} = -\frac{\sqrt{2}}{2} \Rightarrow \sec \frac{5\pi}{4} = -\sqrt{2} \]
Step 3: Apply inverse secant
\[ \sec^{-1}(-\sqrt{2}) \]
The principal value range of \( \sec^{-1}x \) is:
\[ [0, \pi] \setminus \left\{\frac{\pi}{2}\right\} \]
Step 4: Find the correct angle
\[ \sec \theta = -\sqrt{2} \Rightarrow \cos \theta = -\frac{\sqrt{2}}{2} \]
In the interval \( [0, \pi] \), this occurs at:
\[ \theta = \frac{3\pi}{4} \]
Final Answer
\[ \boxed{\frac{3\pi}{4}} \]