Problem
Evaluate: \( \csc^{-1}(\csc \frac{11\pi}{6}) \)
Solution
We know that:
\[ \sin \frac{11\pi}{6} = -\frac{1}{2} \]
So,
\[ \csc \frac{11\pi}{6} = \frac{1}{\sin \frac{11\pi}{6}} = -2 \]
Thus the expression becomes:
\[ \csc^{-1}(-2) \]
Recall the principal value range of \( \csc^{-1} x \):
\[ \left[-\frac{\pi}{2}, 0\right) \cup \left(0, \frac{\pi}{2}\right] \]
Since the value is negative, the angle must lie in:
\[ \left[-\frac{\pi}{2}, 0\right) \]
We know that:
\[ \csc\left(-\frac{\pi}{6}\right) = -2 \]
And \( -\frac{\pi}{6} \) lies in the principal value range.
Final Answer
\[ \boxed{-\frac{\pi}{6}} \]