Check all that apply5/3 is the reference angle for:

2pi/3

15pi/3

7pi/3

19pi/3

To identify if 5/3 is the reference angle for the given angles, we need to understand what a reference angle is. A reference angle is the acute angle formed between the terminal side of an angle in standard position (whose initial side is on the positive x-axis) and the x-axis.

To find the reference angle for an angle, follow these steps:
1. Identify the angle in standard position.
2. Determine which quadrant the angle lies in.
3. If the angle is in the first or second quadrant, the reference angle is the angle itself.
4. If the angle is in the third or fourth quadrant, subtract the angle from 180° (π radians) to find the reference angle.

Let's apply these steps to each given angle:

1. 2pi/3:
- This angle is in the second quadrant.
- Since it is in the second quadrant, the reference angle is equal to 180° (π radians) minus the given angle.
- Reference angle = 180° - 120° = 60° (π/3 radians).
- The reference angle for 2pi/3 is not 5/3.

2. 15pi/3:
- This angle is equivalent to 5π radians.
- Since 5π radians lies on the negative x-axis, it can be considered in the third or fourth quadrant.
- We subtract 5π radians from 180° (π radians) to find the reference angle.
- Reference angle = 180° - 225° = -45° (-π/4 radians).
- The reference angle for 15pi/3 is not 5/3.

3. 7pi/3:
- This angle is equivalent to 14π/3 radians.
- Again, since 14π/3 radians lies on the negative x-axis, it can be considered in the third or fourth quadrant.
- Subtracting 14π/3 radians from 180° (π radians) results in the reference angle.
- Reference angle = 180° - 420° = -240° (-4π radians).
- The reference angle for 7pi/3 is not 5/3.

4. 19pi/3:
- This angle is equivalent to 6π + π/3 radians.
- The angle lies on the positive x-axis as it is in the fourth quadrant.
- Therefore, the reference angle is the angle itself.
- The reference angle for 19pi/3 is 5/3.

Based on the calculations, the only angle for which the reference angle is 5/3 is 19pi/3.