How to Find Co-terminal Angles and Reference Angles
Read,5 minutes
Co-terminal Angles: What Are They?
Angles with the same starting side and a common terminal side are co-terminal angles . Despite having various values, these angles are in a standard position. They share the same sides, are in the same quadrant, and have the same vertices. The terminal sides coincide at the same angle, whether the angles are rotated clockwise or counterclockwise. An angle measures a ray's rotation around its starting point. The starting side of that angle refers to the original ray, while the terminal side denotes the ray's final position upon rotation.
Think about 65°. Due to its terminal side in the first quadrant, it occupies its standard position there. View the illustration.
- 65° returns to its terminal side at 425°, following a complete circle counterclockwise. In the first quadrant, 425° and 65° coincide.
- On a complete clockwise revolution, 65° again reaches its terminal side at −295°. In the first quadrant, −295° and 65° coincide.
As a result, the angles 425° and −295° are co-terminal of 65°.
The formula for Co-terminal Angles
Whether an angle (θ) is expressed in terms of degrees or radians, the following formula can help you determine its co-terminal angles:
- Degrees: θ ± 360n
- Radians: θ ± 2πn
In the above formula, 360n means a number that is a multiple of 360, and n is an integer that shows how many times the coordinate plane has been turned.
As a result, we may say that the following angles are all coterminal: 65°, −295°, 425°, −655°, 785°, etc. They only differ by a certain number of whole circles. According to this definition, "two angles are said to be co-terminal if the difference between the angles is a multiple of 360° (or 2π, if the angle is in terms of radians)".
Example1
Find the positive and negative coterminal angles of 3π4
Solution:
- 3π4 + 2π =3π4 + 8π4 = 11π4
- 3π4 − 2π =3π4 − 8π4 = −5π4
Example2
Find the positive and negative co-terminal angles of 59°.
Solution:
- 59° + 360° = 419°
- 59° − 360° = −301°
Positive and Negative Co-terminal Angles
Positive or negative co-terminal angles are also possible. we found that 419° and −301° are the co-terminal angles of 59°. Here,
- 419° is the coterminal angle of 59° and that is positive.
- −301° The negative co-terminal angle of is 59°.
θ ± 360n, where n takes a positive number for counterclockwise rotations and a negative value for clockwise rotations. In order to get positive or negative co-terminal angles, we must select whether to add or subtract multiples of 360° (or 2π).
Reference Angles
The reference angle for every angle is the angle between 0° and 90°. The quadrant of the terminal side determines the reference angle.
The procedures for determining an angle's reference angle
- First, its co-terminal angle must be calculated, which ranges from 0° to 360°.
- The coterminal angle's quadrant is then visible.
- The reference angle is the same as our given angle if the terminal side is in the first quadrant (0° to 90°). For instance, if the angle is 40° degrees, the reference angle is 40° degrees.
- The reference angle is 180° − given angle if the terminal side is in the second quadrant (90° to 180° degrees). For instance, if the angle is 170°, the reference angle is 180° – 170° = 10°.
- The reference angle is (given angle − 180°) if the terminal side is in the third quadrant (180° to 270°). For instance, if an angle is 195°, its reference angle is 195° – 180°, which equals 15°.
- The reference angle is (360° − provided angle) if the terminal side is in the fourth quadrant (270° to 360°). For instance, if the angle is 305°, the reference angle is 360° – 305° = 55°.
Example:
Find the reference angle of 510°.
Solution:
Find the co-terminal angle of 510°. 510° − 360° = 150° is the co-terminal angle.
The second quadrant contains the terminal side. The reference angle is thus 180° – 150° = 30°.
As a result, 30° is the reference angle for 510°.
Summary
- Co-terminal angles are equal angles.
- To find a co-terminal of an angle, add or subtract 360 degrees (or 2π for radians) to the given angle.
- Reference angle is the smallest angle that you can make from the terminal side of an angle with the x−axis.
Free printable Worksheets
Exercises for Finding Co-terminal Angles and Reference Angles
1) Find the co-terminal angle: 220° =
2) Find the co-terminal angle: 115° =
3) Find the co-terminal angle: 5π4 =
4) Find the co-terminal angle: 13π20 =
5) Find the co-terminal angle: 9π14 =
6) Find the reference angle: 640° =
7) Find the reference angle: 580° =
8) Find the reference angle: 420° =
9) Find the reference angle: 19π8 =
10) Find the reference angle: 27π7 =
1) Find the co-terminal angle: 220° =
- Positive co-terminal: 220° = 220° + 360° = 580°
- Negative co-terminal: 220° = 220° − 360° = −140°
2) Find the co-terminal angle: 115° =
- Positive co-terminal: 115° = 115° + 360° = 475°
- Negative co-terminal: 115° = 115° − 360° = −245°
3) Find the co-terminal angle: 5π4 =
- Positive co-terminal: 5π4 =5π4 + 2π = 13π4
- Negative co-terminal: 5π4 =5π4 − 2π = −3π4
4) Find the co-terminal angle: 13π20 =
- Positive co-terminal: 13π20 =13π20 + 2π = 53π20
- Negative co-terminal: 13π20 =13π20 − 2π = −27π4
5) Find the co-terminal angle: 9π14 =
- Positive co-terminal: 9π14 =9π14 + 2π = 27π14
- Negative co-terminal: 9π14 =9π14 − 2π = −21π14
6) Find the reference angle: 640° =
Co-terminal angle: 640° = 640° − 360° = 280° ⇒ fourth quadrant
⇒ reference angle = 360° − 280° = 80°
7) Find the reference angle: 580° =
Co-terminal angle: 580° = 580° − 360° = 220° ⇒ third quadrant
⇒ reference angle = 220° − 180° = 40°
8) Find the reference angle: 420° =
Co-terminal angle: 420° = 420° − 360° = 60° ⇒ first quadrant
⇒ reference angle = 60°
9) Find the reference angle: 19π8 =
Co-terminal angle: 14π8 = 19π8 − 2π = 3π8 ⇒ first quadrant
⇒ reference angle = 3π8
10) Find the reference angle: 27π7 =
Co-terminal angle: 27π7 = 27π7 − 2π = 13π7 ⇒ fourth quadrant
⇒ reference angle = 2π − 13π7 = π7
Finding Co-terminal Angles and Reference Angles Practice Quiz