Problem
Evaluate: \( \csc^{-1}(\csc(-\frac{9\pi}{4})) \)
Solution
First, use periodicity of sine:
\[ -\frac{9\pi}{4} = -2\pi – \frac{\pi}{4} \]
Since sine is periodic with period \(2\pi\):
\[ \sin\left(-\frac{9\pi}{4}\right) = \sin\left(-\frac{\pi}{4}\right) \]
Now,
\[ \sin\left(-\frac{\pi}{4}\right) = -\frac{1}{\sqrt{2}} \]
So,
\[ \csc\left(-\frac{9\pi}{4}\right) = \frac{1}{\sin\left(-\frac{9\pi}{4}\right)} = -\sqrt{2} \]
Thus the expression becomes:
\[ \csc^{-1}(-\sqrt{2}) \]
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:
\[ \csc\left(-\frac{\pi}{4}\right) = -\sqrt{2} \]
And \( -\frac{\pi}{4} \) lies in the principal value range.
Final Answer
\[ \boxed{-\frac{\pi}{4}} \]