Find A Positive Angle That Is Coterminal Calculator

Coterminal angles are angles that share the same initial and terminal sides. They differ by integer multiples of 360° (or 2π radians). This calculator helps you find a positive angle that is coterminal to any given angle.

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 of 360° (or 2π radians).

For example, 30° and 390° are coterminal because 390° - 360° = 30°. Similarly, -45° and 315° are coterminal because -45° + 360° = 315°.

Coterminal angles are useful in trigonometry, navigation, and engineering where periodic functions like sine and cosine repeat every 360°.

How to Find Coterminal Angles

To find a positive coterminal angle for any given angle θ:

If θ is positive and less than 360°, it is already in the standard range.
If θ is positive and greater than or equal to 360°, subtract 360° repeatedly until the result is less than 360°.
If θ is negative, add 360° repeatedly until the result is positive and less than 360°.

This process ensures you find the equivalent positive angle between 0° and 360°.

Coterminal angle = θ mod 360° (for positive θ) Coterminal angle = (θ + 360°) mod 360° (for negative θ)

Calculator Usage

Use the calculator on the right to find a positive coterminal angle for any given angle. Simply enter the angle in degrees and click "Calculate".

The calculator will display the positive coterminal angle and explain the result. You can also view a visualization of the angle on the unit circle.

Examples

Example 1: Positive Angle Less Than 360°

Given angle: 45°

Since 45° is already between 0° and 360°, the coterminal angle is 45°.

Example 2: Positive Angle Greater Than 360°

Given angle: 405°

Subtract 360°: 405° - 360° = 45°

The positive coterminal angle is 45°.

Example 3: Negative Angle

Given angle: -90°

Add 360°: -90° + 360° = 270°

The positive coterminal angle is 270°.

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 on a straight line.

Can coterminal angles be negative?

Yes, coterminal angles can be negative. However, the calculator finds a positive equivalent by adding or subtracting full rotations.

How are coterminal angles used in real life?

Coterminal angles are used in navigation, engineering, and physics where periodic functions like sine and cosine repeat every 360°.