Coterminal Angle Calculator Degrees

This coterminal angle calculator helps you find all angles that share the same terminal side as a given angle in degrees. Coterminal angles are essential in trigonometry and navigation, where understanding angle relationships is crucial.

What is a Coterminal Angle?

Coterminal angles are angles that share the same terminal side when drawn in standard position. In other words, they can be reached by adding or subtracting full rotations (360°) to the original angle.

For example, 45° and 405° are coterminal because 405° = 45° + 360°. Similarly, -315° and 45° are coterminal because -315° + 360° = 45°.

Coterminal angles are always separated by integer multiples of 360°. This means you can find infinitely many coterminal angles for any given angle.

How to Find Coterminal Angles

To find coterminal angles, you can add or subtract 360° (or any multiple of 360°) to the original angle. This process can be repeated to find multiple coterminal angles.

Step-by-Step Process

Start with the original angle in degrees.
Add 360° to find the next positive coterminal angle.
Subtract 360° to find the next negative coterminal angle.
Repeat the process to find more coterminal angles.

For example, if the original angle is 60°:

60° + 360° = 420°
60° - 360° = -300°
60° + 720° = 780°
60° - 720° = -660°

Coterminal Angle Formula

The general formula to find coterminal angles is:

θ_coterminal = θ + (360° × n)

Where:

θ is the original angle in degrees
n is any integer (positive, negative, or zero)

This formula allows you to find all coterminal angles by substituting different integer values for n.

Examples

Example 1: Finding Coterminal Angles for 90°

Using the formula θ_coterminal = 90° + (360° × n):

For n = 1: 90° + 360° = 450°
For n = -1: 90° - 360° = -270°
For n = 2: 90° + 720° = 810°

Example 2: Finding Coterminal Angles for -45°

Using the formula θ_coterminal = -45° + (360° × n):

For n = 1: -45° + 360° = 315°
For n = -1: -45° - 360° = -405°
For n = 2: -45° + 720° = 675°

FAQ

What is the difference between coterminal and reference angles?: Coterminal angles share the same terminal side, while reference angles are the smallest positive angle that an angle makes with the x-axis in standard position.
Can coterminal angles be negative?: Yes, coterminal angles can be negative. For example, -90° and 270° are coterminal because -90° + 360° = 270°.
How many coterminal angles are there for any given angle?: There are infinitely many coterminal angles for any given angle, as you can keep adding or subtracting full rotations (360°).
Are coterminal angles used in real-world applications?: Yes, coterminal angles are used in navigation, engineering, and physics to simplify angle calculations and understand periodic behavior.