Smallest Positive Coterminal Angle Calculator

Coterminal angles are angles that share the same initial and terminal sides. They differ by integer multiples of 360°. This calculator helps you find the smallest positive coterminal angle for any given angle.

What is a Coterminal Angle?

Coterminal angles are angles that have the same terminal side. They can be found by adding or subtracting full rotations (360°) to an angle. For example, 45° and 405° are coterminal because 405° - 360° = 45°.

Coterminal angles are used in trigonometry, navigation, and engineering to simplify angle calculations and represent the same direction.

Key Properties of Coterminal Angles

All coterminal angles share the same terminal side
They differ by integer multiples of 360°
They have the same sine and cosine values
They can be positive or negative

How to Find the Smallest Positive Coterminal Angle

The smallest positive coterminal angle is found by adding or subtracting full rotations (360°) until the angle is between 0° and 360°.

For any angle θ:

Smallest positive coterminal angle = θ mod 360°

If the result is negative, add 360° to get the positive equivalent

Step-by-Step Calculation

Enter your angle in the calculator
Select the angle unit (degrees or radians)
Click "Calculate" to find the smallest positive coterminal angle
Review the result and visualization

Example Calculation

Find the smallest positive coterminal angle for -100°:

-100° mod 360° = -100° (negative result)
Add 360°: -100° + 360° = 260°
260° is between 0° and 360°, so it's the smallest positive coterminal angle

Applications of Coterminal Angles

Coterminal angles are used in various fields including:

Trigonometry: Simplifying angle calculations
Navigation: Determining directions
Engineering: Designing rotating systems
Physics: Analyzing circular motion
Computer Graphics: Creating animations

Understanding coterminal angles helps in solving problems involving periodic functions and rotational systems.

FAQ

What is the difference between coterminal and reference angles?: Coterminal angles share the same terminal side and differ by full rotations. Reference angles are the smallest positive angles formed with the x-axis, typically between 0° and 90°.
Can coterminal angles be negative?: Yes, coterminal angles can be negative. The smallest positive coterminal angle is always between 0° and 360°.
How do I find coterminal angles in radians?: Use the same method as with degrees, but use 2π radians instead of 360°. The formula is θ mod 2π.
Are all coterminal angles equal?: No, coterminal angles differ by integer multiples of 360° (or 2π radians). They share the same terminal side but can have different measures.