Find The Smallest Positive Coterminal Angle Calculator

What Are Coterminal Angles?

Coterminal angles are angles that have the same terminal side when drawn in standard position. In other words, they can be reached by adding or subtracting full rotations (360°) to an initial angle.

For example, 45° and 405° are coterminal because 405° - 360° = 45°. Similarly, -225° and 135° are coterminal because -225° + 360° = 135°.

The smallest positive coterminal angle is particularly useful in trigonometry and navigation, where angles are often measured from a reference position.

How to Find Coterminal Angles

To find coterminal angles, you can use the following formula:

The smallest positive coterminal angle occurs when n is chosen such that θ_coterminal is between 0° and 360°.

Steps to Find the Smallest Positive Coterminal Angle

1. Start with the original angle θ.
2. If θ is positive and less than 360°, it is already the smallest positive coterminal angle.
3. If θ is negative, add 360° repeatedly until the result is positive.
4. If θ is greater than 360°, subtract 360° repeatedly until the result is less than 360°.

This process ensures you find the smallest positive angle that is coterminal with the original angle.

Using the Calculator

Our calculator makes finding the smallest positive coterminal angle quick and easy. Simply:

1. Enter the original angle in degrees in the input field.
2. Click "Calculate" to find the smallest positive coterminal angle.
3. View the result and see a visualization of the angle on the unit circle.

The calculator handles all cases, including negative angles and angles greater than 360°.

Examples

Example 1: Positive Angle Less Than 360°

Original angle: 45°

Smallest positive coterminal angle: 45° (no calculation needed)

Example 2: Negative Angle

Original angle: -225°

Calculation: -225° + 360° = 135°

Smallest positive coterminal angle: 135°

Example 3: Angle Greater Than 360°

Original angle: 405°

Calculation: 405° - 360° = 45°

Smallest positive coterminal angle: 45°