Find the reference angle for a rotation of start fraction, 13, pi, divided by, 7, end fraction 
7
13π
​	
 .

To find the reference angle for the rotation of \( \frac{13\pi}{7} \), follow these steps:

### Step 1: Understand the concept of reference angles.
The reference angle is the smallest positive acute angle that the terminal side of the given angle makes with the x-axis. For angles measured in radians, the reference angle depends on which quadrant the terminal side lies in.

### Step 2: Reduce the angle to an equivalent angle between \( 0 \) and \( 2\pi \).
Given \( \frac{13\pi}{7} \), the first thing is to reduce it within the interval \( [0, 2\pi] \).
- Since \( 2\pi = \frac{14\pi}{7} \), subtract \( 2\pi \) from \( \frac{13\pi}{7} \) to find the coterminal angle:
  \[
  \frac{13\pi}{7} - 2\pi = \frac{13\pi}{7} - \frac{14\pi}{7} = -\frac{\pi}{7}
  \]
- A negative angle implies clockwise rotation, so add \( 2\pi \) to bring it into the positive interval:
  \[
  -\frac{\pi}{7} + 2\pi = -\frac{\pi}{7} + \frac{14\pi}{7} = \frac{13\pi}{7}
  \]
Thus, the equivalent angle is \( \frac{\pi}{7} \).

### Step 3: Identify the quadrant and find the reference angle.
The reduced angle \( \frac{\pi}{7} \) is in the first quadrant, so it is already the reference angle.

### Final Answer: The reference angle is \( \frac{\pi}{7} \).

Would you like more details or clarifications? Here are some related questions:

1. How do you convert between radians and degrees?
2. What is the significance of reference angles in trigonometry?
3. How do you find the reference angle for angles in the second, third, and fourth quadrants?
4. How do coterminal angles help in determining reference angles?
5. How can negative angles be interpreted geometrically?

**Tip:** To quickly reduce any angle larger than \( 2\pi \) (or \( 360^\circ \)), subtract multiples of \( 2\pi \) (or \( 360^\circ \)) to find a coterminal angle.

Find the reference angle for a rotation of 13π/7.

Discover how to find the reference angle for a rotation of 13π/7. This solution explains key trigonometric concepts like reference angles and coterminal angles, with clear step-by-step instructions. Ideal for students in Grades 10-12.

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