Positive Coterminal Angle Calculator

This positive coterminal angle calculator helps you find all positive angles that are coterminal with a given angle. Coterminal angles share the same terminal side but differ by full rotations (360° or 2π radians).

What are Coterminal Angles?

Coterminal angles are angles that share the same terminal side when drawn in standard position. They differ by integer multiples of 360° (for degrees) or 2π radians (for radians).

For example, 45° and 405° are coterminal because 405° = 45° + 360° × 1. Similarly, -90° and 270° are coterminal because 270° = -90° + 360° × 1.

Key Properties

Coterminal angles have the same sine and cosine values
They differ by full rotations (360° or 2π radians)
All positive coterminal angles are greater than 0° and less than 360°

How to Find Positive Coterminal Angles

To find positive coterminal angles for a given angle θ:

If θ is positive and less than 360°, it is already in the positive range
If θ is negative, add 360° repeatedly until you get a positive angle
If θ is greater than 360°, subtract 360° repeatedly until you get an angle between 0° and 360°

Formula

For any angle θ, the positive coterminal angle θ' can be found using:

θ' = θ mod 360°

If θ' is negative, add 360° to get the positive equivalent.

This calculator implements this logic to find all positive coterminal angles for your input.

Examples

Example 1: Positive Angle Less Than 360°

Input: 45°

Result: 45° (already positive and less than 360°)

Example 2: Negative Angle

Input: -90°

Calculation: -90° + 360° = 270°

Result: 270°

Example 3: Angle Greater Than 360°

Input: 405°

Calculation: 405° - 360° = 45°

Result: 45°

FAQ

What is the difference between coterminal and congruent angles?

Congruent angles are exactly equal in measure, while coterminal angles share the same terminal side but differ by full rotations. For example, 45° and 405° are coterminal but not congruent.

How many positive coterminal angles exist for any given angle?

There is exactly one positive coterminal angle between 0° and 360° for any given angle. This is the angle you get after applying the modulo operation with 360°.

Can coterminal angles be negative?

Yes, coterminal angles can be negative. However, the positive coterminal angle calculator specifically finds the equivalent positive angle between 0° and 360°.