Sure! Let's work through an example similar to $\csc(-\frac{3\pi}{2})$ with a different angle. I'll use $\csc(-\frac{\pi}{4})$.

Example: Find $\csc\left(-\frac{\pi}{4}\right)$.

Step 1: Recall the definition of cosecant.

$\csc(\theta) = \frac{1}{\sin(\theta)}$ So, $\csc\left(-\frac{\pi}{4}\right) = \frac{1}{\sin\left(-\frac{\pi}{4}\right)}$.

Step 2: Find $\sin\left(-\frac{\pi}{4}\right)$.

Using the fact that sine is an odd function: $\sin(-\theta) = -\sin(\theta)$ Thus, $\sin\left(-\frac{\pi}{4}\right) = -\sin\left(\frac{\pi}{4}\right).$ We know $\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$, so: $\sin\left(-\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2}.$

Step 3: Calculate $\csc\left(-\frac{\pi}{4}\right)$.

$\csc\left(-\frac{\pi}{4}\right) = \frac{1}{\sin\left(-\frac{\pi}{4}\right)} = \frac{1}{-\frac{\sqrt{2}}{2}} = -\frac{2}{\sqrt{2}} = -\sqrt{2}.$

Thus, $\csc\left(-\frac{\pi}{4}\right) = -\sqrt{2}$.

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Would you like more details on any of these steps or another example?

Here are five related questions you could try:

1. What is $\csc(\frac{\pi}{6})$?
2. How does $\csc(\theta)$ change when $\theta$ is negative?
3. Find $\csc(\frac{\pi}{3})$.
4. Can you calculate $\csc(-\pi)$?
5. Why is $\csc(0)$ undefined?

Tip: Remember that sine and cosecant are reciprocal functions, so if $\sin(\theta)$ is small, $\csc(\theta)$ will be large!