Coterminal Angles Degrees Calculator

Coterminal angles are angles that share the same initial and terminal sides. They differ by integer multiples of 360°. This calculator helps you find all coterminal angles for any given angle in degrees.

What Are Coterminal Angles?

Coterminal angles are angles in standard position (with their vertex at the origin and initial side along the positive x-axis) that have the same terminal side. They can be found by adding or subtracting full rotations (360°) to the original angle.

For example, 45° and 405° are coterminal because 405° - 360° = 45°. Similarly, -270° and 90° are coterminal because -270° + 360° = 90°.

How to Find Coterminal Angles

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

Start with your original angle in degrees.
Add 360° to get the next positive coterminal angle.
Subtract 360° to get the next negative coterminal angle.
Repeat the process to find as many coterminal angles as needed.

Coterminal Angles Formula

The general formula for finding 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 infinitely many coterminal angles by choosing different values for n.

Examples of Coterminal Angles

Example 1: Positive Angle

Find three coterminal angles for 60°.

60° + 360° × 1 = 420°
60° + 360° × (-1) = -300°
60° + 360° × 2 = 780°

So, 60°, 420°, -300°, and 780° are all coterminal angles.

Example 2: Negative Angle

Find two coterminal angles for -120°.

-120° + 360° × 1 = 240°
-120° + 360° × (-2) = -840°

So, -120°, 240°, and -840° are all coterminal angles.

FAQ

What is the difference between coterminal and supplementary angles?: Coterminal angles share the same terminal side and differ by full rotations (360°). Supplementary angles add up to 180° and are not necessarily coterminal.
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 360° to find new ones.
Are all angles coterminal with themselves?: Yes, any angle is coterminal with itself because adding 360° × 0 = 0° to the angle gives the same angle.
How do coterminal angles relate to trigonometric functions?: Coterminal angles have identical trigonometric function values because they share the same terminal side. For example, sin(45°) = sin(405°).