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### **Example 2: Recognizing Coterminal Angles**
Determine whether the following pairs of angles are coterminal:

Coterminal angles share the same initial and terminal sides. Two angles are coterminal if the difference between them is a multiple of \(360^\circ\).

1. **(a) \( \alpha = 120^\circ, \beta = -180^\circ \):**
   - To determine if they are coterminal, calculate the difference between the two angles:
     \[
     120^\circ - (-180^\circ) = 120^\circ + 180^\circ = 300^\circ.
     \]
   - Since \(300^\circ\) is not a multiple of \(360^\circ\), these angles are **not coterminal**.

2. **(b) \( \alpha = 20^\circ, \beta = 740^\circ \):**
   - Find the coterminal angle of \(740^\circ\) by subtracting multiples of \(360^\circ\) to bring it within the range \(0^\circ\) to \(360^\circ\):
     \[
     740^\circ - 360^\circ = 380^\circ,
     \]
     \[
     380^\circ - 360^\circ = 20^\circ.
     \]
   - The reduced angle for \(740^\circ\) is \(20^\circ\). Since this matches \(\alpha\), the angles **are coterminal**.

### **Example 3: Finding Measures of Coterminal Angles**
Determine the angle of the smallest possible positive measure that is coterminal with each of the following angles.

1. **(a) \(830^\circ\):**
   - To find the coterminal angle, subtract \(360^\circ\) repeatedly until the angle is between \(0^\circ\) and \(360^\circ\):
     \[
     830^\circ - 360^\circ = 470^\circ,
     \]
     \[
     470^\circ - 360^\circ = 110^\circ.
     \]
   - The smallest positive coterminal angle is \(110^\circ\).

2. **(b) \(-520^\circ\):**
   - Since the angle is negative, add \(360^\circ\) repeatedly until the angle is between \(0^\circ\) and \(360^\circ\):
     \[
     -520^\circ + 360^\circ = -160^\circ,
     \]
     \[
     -160^\circ + 360^\circ = 200^\circ.
     \]
   - The smallest positive coterminal angle is \(200^\circ\).

### Summary of Answers
1. **Example 2:**
   - (a) Not coterminal.
   - (b) Coterminal.

2. **Example 3:**
   - (a) \(110^\circ\).
   - (b) \(200^\circ\).

Would you like more details or have any other questions?

### 5 Related Questions to Expand Understanding:
1. How do you find a negative coterminal angle for a given positive angle?
2. Can two angles be coterminal if one is in radians and the other in degrees?
3. How do coterminal angles relate to the unit circle?
4. What is the general formula to find all coterminal angles of a given angle?
5. What is the significance of coterminal angles in trigonometric functions?

### Tip:
To quickly find a coterminal angle between \(0^\circ\) and \(360^\circ\), keep adding or subtracting \(360^\circ\) until the angle falls within that range.

Determine whether the following pairs of angles are coterminal and find the smallest positive coterminal angles for given angles.

Learn how to determine coterminal angles and find the smallest positive coterminal angles. This guide walks through examples of identifying coterminal angles by comparing the difference with 360°, suitable for high school students.

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