Coterminal Angles Calculator Degrees

Coterminal angles are angles that share the same initial and terminal sides. They differ by integer multiples of 360 degrees. 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 that have the same terminal side. In other words, they can be rotated by full circles (360 degrees) and still point in the same direction.

For example, 30° and 390° are coterminal because you can add 360° to 30° to get 390°. Similarly, -330° is coterminal with 30° because -330° + 360° = 30°.

Coterminal angles are often used in trigonometry, navigation, and engineering to simplify angle calculations and comparisons.

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 will give you an equivalent angle that shares the same terminal side.

For example:

For 45°, one coterminal angle is 45° + 360° = 405°
For -120°, one coterminal angle is -120° + 360° = 240°

You can find infinitely many coterminal angles by continuing to add or subtract 360°.

Coterminal Angles Formula

Formula

For any angle θ in degrees, coterminal angles can be calculated as:

θ + 360° × n, where n is any integer (positive, negative, or zero)

This formula allows you to generate all possible coterminal angles for any given angle by substituting different integer values for n.

Examples

Example 1: Finding Coterminal Angles for 60°

Using the formula θ + 360° × n:

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

All these angles (420°, -300°, 780°, etc.) are coterminal with 60°.

Example 2: Finding Coterminal Angles for -210°

Using the formula θ + 360° × n:

For n = 1: -210° + 360° × 1 = 150°
For n = -1: -210° + 360° × (-1) = -570°
For n = 2: -210° + 360° × 2 = 510°

All these angles (150°, -570°, 510°, etc.) are coterminal with -210°.

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 often used in geometry to find complementary angles.
Can coterminal angles be negative?: Yes, coterminal angles can be negative. For example, -30° and 330° are coterminal because -30° + 360° = 330°.
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 generate new coterminal angles.
Are coterminal angles used in real-world applications?: Yes, coterminal angles are used in navigation, engineering, and physics to simplify angle calculations and comparisons in circular motion and periodic phenomena.
How do I find the reference angle for a coterminal angle?: The reference angle is the smallest positive acute angle that the terminal side of the angle makes with the x-axis. For any angle θ, you can find the reference angle by taking θ modulo 360° and then adjusting for the quadrant.