Smallest Positive Angle Coterminal Calculator

Coterminal angles are angles that share the same terminal side when drawn in standard position. This calculator helps you find the smallest positive angle coterminal with any given angle. Understanding coterminal angles is essential in trigonometry, navigation, and various engineering applications.

What is a Coterminal Angle?

Coterminal angles are angles that have the same terminal side when drawn in standard position. In other words, they differ by a full rotation (360° or 2π radians) from each other. For example, 45° and 405° are coterminal because 405° - 360° = 45°.

Coterminal angles are useful in trigonometry because they allow us to simplify calculations by working with angles within the first rotation (0° to 360° or 0 to 2π radians).

How to Find Coterminal Angles

To find coterminal angles, you can add or subtract full rotations (360° or 2π radians) from the original angle. The smallest positive coterminal angle is found by:

If the angle is positive: angle % 360° (for degrees) or angle % (2π) (for radians)
If the angle is negative: 360° - (|angle| % 360°) (for degrees) or 2π - (|angle| % 2π) (for radians)

This calculation ensures you get 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 enter your angle in degrees or radians, and the calculator will display the result along with a visual representation.

The calculator uses the following formula:

If angle ≥ 0:
smallestPositive = angle % 360° (degrees)
smallestPositive = angle % (2π) (radians)

If angle < 0:
smallestPositive = 360° - (|angle| % 360°) (degrees)
smallestPositive = 2π - (|angle| % 2π) (radians)

The calculator also provides a chart showing the relationship between the original angle and its coterminal angles.

Examples

Example 1: Positive Angle in Degrees

Find the smallest positive angle coterminal with 405°.

Calculation: 405° % 360° = 45°

Result: 45° is the smallest positive angle coterminal with 405°.

Example 2: Negative Angle in Radians

Find the smallest positive angle coterminal with -π/2 radians.

Calculation: 2π - (|-π/2| % 2π) = 2π - (π/2) = 3π/2

Result: 3π/2 radians is the smallest positive angle coterminal with -π/2 radians.

FAQ

What is the difference between coterminal and equivalent angles?: Coterminal angles share the same terminal side, while equivalent angles are identical in measure. All coterminal angles are equivalent if they differ by full rotations, but not all equivalent angles are coterminal.
Can coterminal angles be negative?: Yes, coterminal angles can be negative. However, the smallest positive coterminal angle is always between 0° and 360° (or 0 and 2π radians).
How are coterminal angles used in real life?: Coterminal angles are used in navigation, engineering, and physics to simplify calculations involving rotational motion and periodic functions.
What is the smallest positive angle coterminal with 0°?: The smallest positive angle coterminal with 0° is 0° itself, since 0° is already in the smallest positive range.