Find The Positive and Negative Coterminal Angle Calculator

This calculator helps you find both positive and negative coterminal angles for any given angle. Coterminal angles share the same terminal side but differ by full rotations (360° or 2π radians). Understanding coterminal angles is essential in trigonometry, navigation, and engineering applications.

What are Coterminal Angles?

Coterminal angles are angles that share the same terminal side when drawn in standard position. This means they differ by integer multiples of 360° (for degrees) or 2π radians (for radians). Coterminal angles have the same sine and cosine values because they terminate at the same point on the unit circle.

The key properties of coterminal angles are:

They have the same trigonometric values (sin, cos, tan)
They differ by full rotations (360° or 2π)
They can be positive or negative
They are periodic with a period of 360° or 2π

Coterminal angles are fundamental in trigonometry and are used in solving problems involving periodic functions, circular motion, and wave patterns.

How to Find Coterminal Angles

To find coterminal angles, you can add or subtract full rotations (360° or 2π) from the original angle. The general formulas are:

θ_coterminal = θ + n × 360° (for degrees)
θ_coterminal = θ + n × 2π (for radians)

Where:

θ is the original angle
n is any integer (positive or negative)

For positive coterminal angles, you use positive values of n. For negative coterminal angles, you use negative values of n.

Positive Coterminal Angle

A positive coterminal angle is obtained by adding full rotations to the original angle. This results in an angle that is larger than the original but still terminates at the same point on the unit circle.

For example, if you have an angle of 45°, its positive coterminal angles would be:

45° + 360° = 405°
45° + 720° = 765°
45° + 1080° = 1125°

All these angles are coterminal with 45° and share the same trigonometric values.

Negative Coterminal Angle

A negative coterminal angle is obtained by subtracting full rotations from the original angle. This results in an angle that is smaller than the original but still terminates at the same point on the unit circle.

For example, if you have an angle of 45°, its negative coterminal angles would be:

45° - 360° = -315°
45° - 720° = -675°
45° - 1080° = -1035°

All these angles are coterminal with 45° and share the same trigonometric values.

Example Calculations

Let's look at an example to find both positive and negative coterminal angles for 120°.

Positive Coterminal Angle

To find a positive coterminal angle, we add 360° to the original angle:

120° + 360° = 480°

480° is a positive coterminal angle of 120°.

Negative Coterminal Angle

To find a negative coterminal angle, we subtract 360° from the original angle:

120° - 360° = -240°

-240° is a negative coterminal angle of 120°.

Multiple Coterminal Angles

You can find multiple coterminal angles by using different integer values for n:

For n = 1: 120° + 360° = 480°
For n = 2: 120° + 720° = 840°
For n = -1: 120° - 360° = -240°
For n = -2: 120° - 720° = -540°

All these angles (480°, 840°, -240°, -540°) are coterminal with 120°.

FAQ

What is the difference between coterminal and reference angles?: Coterminal angles share the same terminal side but differ by full rotations. Reference angles are the smallest positive angles that terminate on the same ray as the original angle, measured from the x-axis.
How many coterminal angles can an angle have?: An angle has infinitely many coterminal angles because you can keep adding or subtracting full rotations (360° or 2π) to get new coterminal angles.
Are coterminal angles only in degrees or can they be in radians?: Coterminal angles can be in either degrees or radians. The key is that they differ by full rotations (360° or 2π).
Why are coterminal angles important in trigonometry?: Coterminal angles help simplify trigonometric problems by reducing angles to their equivalent within one full rotation (0° to 360° or 0 to 2π).
Can coterminal angles be negative?: Yes, coterminal angles can be negative. They represent angles that are measured in the clockwise direction from the positive x-axis.