Find the reference angle for each given angle in the coordinate plane.

I'll solve this step by step, finding the reference angle for each of the 6 problems shown in the image.

First, let me recall what a reference angle is: it's the acute angle (between 0° and 90°, or 0 and π/2 radians) formed between the terminal side of the given angle and the x-axis. It's always positive and less than or equal to 90°.

Let me work through each problem one by one:

Problem 1: -230°
- This is a negative angle, so we rotate clockwise from the positive x-axis.
- -230° is equivalent to 360° - 230° = 130° in the counterclockwise direction (or we can think of it as rotating 230° clockwise).
- The terminal side is in Quadrant II (since 90° < 130° < 180°).
- In Quadrant II, the reference angle = 180° - θ = 180° - 130° = 50°.

Problem 2: -25π/18
- First, let's convert this to degrees to better understand its position: (-25π/18) × (180°/π) = -250°.
- -250° is equivalent to 360° - 250° = 110° in the counterclockwise direction.
- The terminal side is in Quadrant II (since 90° < 110° < 180°).
- In Quadrant II, the reference angle = 180° - θ = 180° - 110° = 70°.
- Converting back to radians: 70° × (π/180°) = 7π/18.

Problem 3: -7π/9
- Convert to degrees: (-7π/9) × (180°/π) = -140°.
- -140° is equivalent to 360° - 140° = 220° in the counterclockwise direction.
- The terminal side is in Quadrant III (since 180° < 220° < 270°).
- In Quadrant III, the reference angle = θ - 180° = 220° - 180° = 40°.
- Converting back to radians: 40° × (π/180°) = 2π/9.

Problem 4: -29π/18
- Convert to degrees: (-29π/18) × (180°/π) = -290°.
- -290° is equivalent to 360° - 290° = 70° in the counterclockwise direction.
- The terminal side is in Quadrant I (since 0° < 70° < 90°).
- In Quadrant I, the reference angle = θ = 70°.
- Converting back to radians: 70° × (π/180°) = 7π/18.

Problem 5: 31π/9
- First, let's reduce this angle by subtracting full rotations (2π = 18π/9): 31π/9 - 18π/9 = 13π/9.
- 13π/9 is still greater than 2π? No, 13π/9 ≈ 4.54, while 2π ≈ 6.28, so 13π/9 is between π and 2π.
- Convert 13π/9 to degrees: (13π/9) × (180°/π) = 260°.
- The terminal side is in Quadrant III (since 180° < 260° < 270°).
- In Quadrant III, the reference angle = θ - 180° = 260° - 180° = 80°.
- Converting back to radians: 80° × (π/180°) = 4π/9.

Problem 6: 640°
- First, reduce this angle by subtracting full rotations (360°): 640° - 360° = 280°.
- The terminal side is in Quadrant IV (since 270° < 280° < 360°).
- In Quadrant IV, the reference angle = 360° - θ = 360° - 280° = 80°.

Let me double-check my calculations to ensure accuracy.

The reference angles are:
1) 50°
2) 7π/18
3) 2π/9
4) 7π/18
5) 4π/9
6) 80°