Evaluating inverse trigonometric functions Without using a calculator, evaluate the following expressions.
$\csc^{-1}\left(-1\right)$

Verified step by step guidance

1. Understand that \( \csc^{-1}(x) \) is the inverse cosecant function, which gives the angle \( \theta \) such that \( \csc(\theta) = x \).
2. Recall that \( \csc(\theta) = \frac{1}{\sin(\theta)} \). Therefore, \( \csc^{-1}(-1) \) means we are looking for an angle \( \theta \) where \( \sin(\theta) = -1 \).
3. The sine function \( \sin(\theta) \) equals \(-1\) at specific angles. Consider the unit circle: \( \sin(\theta) = -1 \) at \( \theta = \frac{3\pi}{2} \) (or \( 270^\circ \)).
4. Verify that \( \theta = \frac{3\pi}{2} \) is within the range of the inverse cosecant function. The principal range for \( \csc^{-1}(x) \) is \([-\frac{\pi}{2}, \frac{\pi}{2}] \) excluding \( 0 \), but for negative values, we consider angles in the third and fourth quadrants.
5. Conclude that the angle \( \theta = \frac{3\pi}{2} \) satisfies the condition \( \csc(\theta) = -1 \), and thus \( \csc^{-1}(-1) = \frac{3\pi}{2} \).