Find the positive and negative coterminal angles of -9π/2

To find the positive and negative coterminal angles of \(-\frac{9\pi}{2}\), follow these steps:

### 1. **Understanding Coterminal Angles:**
   Coterminal angles differ by an integer multiple of \(2\pi\). For any angle \(\theta\), coterminal angles can be found using the formula:
   \[
   \theta + 2k\pi
   \]
   where \(k\) is any integer (positive, negative, or zero).

### 2. **Given Angle:**
   \(\theta = -\frac{9\pi}{2}\)

### 3. **Finding the Positive Coterminal Angle:**

   Start by adding \(2\pi\) to \(\theta\) repeatedly until the result is positive.
   \[
   -\frac{9\pi}{2} + 2\pi = -\frac{9\pi}{2} + \frac{4\pi}{2} = -\frac{5\pi}{2}
   \]
   \[
   -\frac{5\pi}{2} + 2\pi = -\frac{5\pi}{2} + \frac{4\pi}{2} = -\frac{\pi}{2}
   \]
   \[
   -\frac{\pi}{2} + 2\pi = -\frac{\pi}{2} + \frac{4\pi}{2} = \frac{3\pi}{2}
   \]

   Thus, the **positive coterminal angle** is \(\frac{3\pi}{2}\).

### 4. **Finding the Negative Coterminal Angle:**

   Subtract \(2\pi\) from \(\theta\) to find another negative coterminal angle.
   \[
   -\frac{9\pi}{2} - 2\pi = -\frac{9\pi}{2} - \frac{4\pi}{2} = -\frac{13\pi}{2}
   \]

   The **negative coterminal angle** is \(-\frac{13\pi}{2}\).

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

Would you like further details or have any questions about this process?

Here are some related questions to deepen your understanding:

1. What are coterminal angles and how do they relate to the unit circle?
2. How do you convert between radians and degrees when working with coterminal angles?
3. Can coterminal angles be larger than \(2\pi\) or less than \(-2\pi\)?
4. How would you find coterminal angles for angles given in degrees instead of radians?
5. What is the significance of \(2\pi\) in the context of angle measurement?

**Tip:** Coterminal angles always share the same terminal side, which means they represent the same direction on the unit circle.

Learn how to find the positive and negative coterminal angles of -9π/2 using trigonometric principles. This problem explores coterminal angles, their definition, and how to calculate them step-by-step. Suitable for advanced high school students.

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