https://file.aiquickdraw.com/mathbot/file/1725505391060cgc62u0x.jpg

The image contains two questions asking for the reference angles of given angles in radians:

1. The reference angle of \( \frac{12\pi}{7} \) radians.
2. The reference angle of \( \frac{35\pi}{8} \) radians.

### To solve these:

#### Step 1: Understanding Reference Angles
A reference angle is the smallest angle made between the terminal side of an angle and the x-axis. It's always positive and lies between \( 0 \) and \( \frac{\pi}{2} \) radians (or between \( 0^\circ \) and \( 90^\circ \)).

#### Step 2: Convert the angles into their equivalent within \( [0, 2\pi] \) range.

**1. For \( \frac{12\pi}{7} \):**
- \( \frac{12\pi}{7} \) radians is greater than \( 2\pi \) (since \( 2\pi = \frac{14\pi}{7} \)).
- To find the coterminal angle, subtract \( 2\pi \) from \( \frac{12\pi}{7} \):
  \[
  \frac{12\pi}{7} - 2\pi = \frac{12\pi}{7} - \frac{14\pi}{7} = -\frac{2\pi}{7}
  \]
- Since the angle is negative, we add \( 2\pi \) to get it positive:
  \[
  -\frac{2\pi}{7} + 2\pi = \frac{12\pi}{7}
  \]
  This angle is now positive within the first quadrant. So, the reference angle is \( \frac{2\pi}{7} \) radians.

**2. For \( \frac{35\pi}{8} \):**
- \( \frac{35\pi}{8} \) radians is greater than \( 2\pi \) (since \( 2\pi = \frac{16\pi}{8} \)).
- To find the coterminal angle, subtract \( 2\pi \) multiple times:
  \[
  \frac{35\pi}{8} - 2\pi = \frac{35\pi}{8} - \frac{16\pi}{8} = \frac{19\pi}{8}
  \]
- Since this is still greater than \( 2\pi \), subtract \( 2\pi \) again:
  \[
  \frac{19\pi}{8} - \frac{16\pi}{8} = \frac{3\pi}{8}
  \]
- Now \( \frac{3\pi}{8} \) lies within \( [0, \pi] \), so it is the reference angle.

### Final Answers:
1. The reference angle for \( \frac{12\pi}{7} \) is \( \frac{2\pi}{7} \) radians.
2. The reference angle for \( \frac{35\pi}{8} \) is \( \frac{3\pi}{8} \) radians.

Would you like further explanation on how to find reference angles? 

Here are 5 related questions to explore this concept further:
1. How do reference angles differ between different quadrants?
2. How do you calculate the reference angle for angles in degrees?
3. What is the reference angle of \( \frac{9\pi}{4} \)?
4. How do reference angles help in solving trigonometric functions?
5. How do coterminal angles relate to reference angles?

**Tip**: When dealing with angles larger than \( 2\pi \) (or 360°), always reduce them by subtracting multiples of \( 2\pi \).

The reference angle of 12π/7 radians is ____. The reference angle of 35π/8 radians is ____.

Learn how to find the reference angles for 12π/7 and 35π/8 radians with a step-by-step solution. This problem focuses on key trigonometric concepts like reference angles in radians. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation