https://file.aiquickdraw.com/mathbot/file/1726625700219y9auiybh.jpg

Let's go through the provided problems step by step.

### Example 2: Recognizing Coterminal Angles
Two angles are coterminal if their difference is a multiple of \( 360^\circ \) (i.e., they differ by \( 360^\circ \) or any multiple of it).

#### (a) \( \alpha = 120^\circ \), \( \beta = -180^\circ \)
To check if these angles are coterminal, calculate the difference between the two angles:

\[
\text{Difference} = 120^\circ - (-180^\circ) = 120^\circ + 180^\circ = 300^\circ
\]

Since \( 300^\circ \) is not a multiple of \( 360^\circ \), these angles are **not coterminal**.

#### (b) \( \alpha = 20^\circ \), \( \beta = 740^\circ \)
Now, calculate the difference between the two angles:

\[
\text{Difference} = 740^\circ - 20^\circ = 720^\circ
\]

Since \( 720^\circ = 2 \times 360^\circ \), these angles are **coterminal** because their difference is a multiple of \( 360^\circ \).

---

### Example 3: Finding Measures of Coterminal Angles
For this example, we need to find the smallest positive coterminal angle. The coterminal angle can be found using the formula:
\[
\theta_{\text{coterminal}} = \theta \mod 360^\circ
\]
If the result is negative, add \( 360^\circ \) to get the smallest positive angle.

#### (a) \( 830^\circ \)
First, calculate the coterminal angle:
\[
830^\circ \mod 360^\circ = 830^\circ - 2 \times 360^\circ = 830^\circ - 720^\circ = 110^\circ
\]
Thus, the smallest positive coterminal angle is \( 110^\circ \).

#### (b) \( -520^\circ \)
Now, calculate the coterminal angle:
\[
-520^\circ \mod 360^\circ = -520^\circ + 2 \times 360^\circ = -520^\circ + 720^\circ = 200^\circ
\]
Thus, the smallest positive coterminal angle is \( 200^\circ \).

---

### Summary:
- Example 2:
  - (a) \( \alpha = 120^\circ \), \( \beta = -180^\circ \): **Not coterminal**
  - (b) \( \alpha = 20^\circ \), \( \beta = 740^\circ \): **Coterminal**
- Example 3:
  - (a) \( 830^\circ \): Smallest positive coterminal angle is \( 110^\circ \)
  - (b) \( -520^\circ \): Smallest positive coterminal angle is \( 200^\circ \)

---

Would you like further clarification on any of these steps or concepts?

Here are 5 related questions to expand your understanding:
1. What is the general formula for finding coterminal angles?
2. Can two negative angles be coterminal?
3. How do you find the negative coterminal angle for a given angle?
4. If an angle is greater than \( 720^\circ \), how would you find its smallest coterminal angle?
5. What are the real-world applications of coterminal angles?

### Tip:
Always remember that coterminal angles are simply angles that share the same terminal side, so you can add or subtract \( 360^\circ \) repeatedly to find an equivalent angle within a specified range.

Determine whether the following pairs of angles are coterminal. a. α = 120°, β = -180° b. α = 20°, β = 740°. Determine the angle of the smallest possible positive measure that is coterminal with each of the following angles: a. 830° b. -520°

Learn how to recognize and calculate coterminal angles with detailed step-by-step solutions. This problem set covers trigonometric concepts and provides examples like determining coterminal angles for α = 120°, β = -180° and finding the smallest positive coterminal angle for 830° and -520°.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation