Find Coterminal Angles in Degrees Calculator

Coterminal angles are angles that share the same terminal side when drawn in standard position. 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 that have the same terminal side when drawn in standard position. In other words, they differ by full rotations (360 degrees) of the unit circle. For example, 30° and 390° are coterminal because 390° - 30° = 360°.

Formula: θ_coterminal = θ + 360° × n, where n is any integer (positive or negative).

Coterminal angles are important in trigonometry because they simplify calculations by allowing you to work with angles within the standard range of 0° to 360°.

How to Find Coterminal Angles

To find coterminal angles, you can use the formula:

θ_coterminal = θ + 360° × n, where n is any integer.

This formula allows you to generate an infinite number of coterminal angles by substituting different integer values for n. For example, if you have an angle of 45°, you can find coterminal angles by adding or subtracting 360°:

45° + 360° × 1 = 405°
45° + 360° × (-1) = -315°
45° + 360° × 2 = 765°

You can also find coterminal angles by reducing the given angle modulo 360° to find the reference angle within the standard range of 0° to 360°.

Using the Calculator

Our calculator makes it easy to find coterminal angles. Simply enter the angle in degrees and click "Calculate". The calculator will display the reference angle and generate a list of coterminal angles.

Note: The calculator shows coterminal angles for n = -2 to n = 2 by default. You can adjust the range if needed.

The calculator also provides a visual representation of the angles on a unit circle chart.

Examples

Let's look at a few examples to understand how coterminal angles work.

Example 1: 60°

For an angle of 60°, the reference angle is 60° itself. Some coterminal angles are:

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

Example 2: -120°

For an angle of -120°, the reference angle is 240° (360° - 120°). Some coterminal angles are:

-120° + 360° × 1 = 240°
-120° + 360° × 2 = 600°
-120° + 360° × (-1) = -480°

FAQ

What is the difference between coterminal and reference angles?: Coterminal angles share the same terminal side and differ by full rotations (360°). Reference angles are the smallest positive angle that an angle makes with the x-axis, always between 0° and 90°.
How many coterminal angles are there for any given angle?: There are infinitely many coterminal angles for any given angle, as you can add or subtract any integer multiple of 360°.
Can coterminal angles be negative?: Yes, coterminal angles can be negative. For example, -45° and 315° are coterminal because -45° + 360° = 315°.
How do coterminal angles relate to the unit circle?: Coterminal angles all terminate at the same point on the unit circle. This means they have the same sine and cosine values.
Are coterminal angles used in real-world applications?: Yes, coterminal angles are used in various real-world applications, including navigation, engineering, and physics, where periodic functions and circular motion are involved.