Find the positive and a negative coterminal angle for each given angle.
-9𝜋 / 2 


To find coterminal angles for a given angle, we add or subtract full rotations of \(2\pi\) (in radians) until we find an angle within the desired range.

Given the angle \(-\frac{9\pi}{2}\), let's find a positive and a negative coterminal angle.

### Step-by-Step Solution

1. **Understanding Coterminal Angles:**
   - Coterminal angles differ by full rotations of \(2\pi\).
   - To find a coterminal angle, we can add or subtract \(2\pi\) repeatedly until we reach the desired range (positive or negative).

2. **Finding a Positive Coterminal Angle:**

   To find a positive coterminal angle, we add \(2\pi\) until the result is positive:
   
   \[
   -\frac{9\pi}{2} + 2\pi
   \]
   
   Since adding \(2\pi\) is the same as adding \(\frac{4\pi}{2}\), we have:
   
   \[
   -\frac{9\pi}{2} + \frac{4\pi}{2} = -\frac{5\pi}{2}
   \]

   This angle is still negative, so we add another \(2\pi\) (or \(\frac{4\pi}{2}\)):

   \[
   -\frac{5\pi}{2} + \frac{4\pi}{2} = -\frac{\pi}{2}
   \]

   This angle is still negative, so we add another \(2\pi\):

   \[
   -\frac{\pi}{2} + \frac{4\pi}{2} = \frac{3\pi}{2}
   \]

   Now, \(\frac{3\pi}{2}\) is positive, so this is our **positive coterminal angle**.

3. **Finding a Negative Coterminal Angle:**

   To find a negative coterminal angle, we subtract \(2\pi\) from the original angle:

   \[
   -\frac{9\pi}{2} - 2\pi
   \]

   Again, subtracting \(2\pi\) is the same as subtracting \(\frac{4\pi}{2}\):

   \[
   -\frac{9\pi}{2} - \frac{4\pi}{2} = -\frac{13\pi}{2}
   \]

   This is a negative angle, so \(-\frac{13\pi}{2}\) is our **negative coterminal angle**.

### Summary

- **Positive Coterminal Angle:** \(\frac{3\pi}{2}\)
- **Negative Coterminal Angle:** \(-\frac{13\pi}{2}\)

Would you like more details, or do you have any questions?

### Related Questions

1. What are coterminal angles for \(3\pi\)?
2. How to convert \(-210^\circ\) to radians and find its coterminal angles?
3. What is the smallest positive coterminal angle of \(-\frac{5\pi}{4}\)?
4. How to find the reference angle for \(-\frac{9\pi}{2}\)?
5. What is the difference between reference angles and coterminal angles?

### Tip

To quickly find coterminal angles in degrees, add or subtract \(360^\circ\); for radians, add or subtract \(2\pi\).

Learn how to find positive and negative coterminal angles for the angle -9𝜋/2 in radians. Step-by-step solution provided, covering concepts of coterminal angles and radians. Suitable for Grades 10-12.

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