can you give me another example of this practice problem: csc(-3pi/2)

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!

Learn how to calculate the cosecant of angles such as -3π/2 with a step-by-step example. This practice problem covers important trigonometric concepts, including the cosecant function, the reciprocal of sine, and the property of odd functions. Perfect for students in Grades 10-12.

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