Problem
Evaluate: \( \csc^{-1}(\csc \frac{13\pi}{6}) \)
Solution
First, reduce the angle:
\[ \frac{13\pi}{6} = 2\pi + \frac{\pi}{6} \]
Using periodicity:
\[ \sin\left(2\pi + \theta\right) = \sin \theta \]
So,
\[ \sin \frac{13\pi}{6} = \sin \frac{\pi}{6} = \frac{1}{2} \]
Thus,
\[ \csc \frac{13\pi}{6} = \frac{1}{\sin \frac{13\pi}{6}} = 2 \]
Now the expression becomes:
\[ \csc^{-1}(2) \]
Recall the principal value range of \( \csc^{-1} x \):
\[ \left[-\frac{\pi}{2}, 0\right) \cup \left(0, \frac{\pi}{2}\right] \]
We need an angle in this range whose cosecant is 2.
We know that:
\[ \csc \frac{\pi}{6} = 2 \]
And \( \frac{\pi}{6} \) lies in the principal value range.
Final Answer
\[ \boxed{\frac{\pi}{6}} \]