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Coterminal Angle Calculator

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The online Coterminal Angle Calculator helps you quickly determine the positive and negative coterminal angles for any given angle. It can also verify whether two angles are coterminal.

What is a Coterminal Angle?

Coterminal angles are angles that share the same initial and terminal sides when drawn in standard position. This means their initial sides start on the positive x-axis, and they end at the same location. For example, 110° and -250° are coterminal angles.

Key definitions:

Initial side: The starting line from which the angle is measured.
Terminal side: The ending line where the angle measurement stops.

Coterminal Angles Diagram

Finding Positive and Negative Coterminal Angles

To calculate coterminal angles, simply add or subtract full rotations (360° for degrees, 2π for radians) from the original angle. Example:

Given angle: α = 14°
Positive coterminal angle: β = α + 360 = 14° + 360° = 374°
Negative coterminal angle: β = α - 360 = 14° - 360° = -346°

Step-by-Step Method to Find Coterminal Angles

If the angle is in radians, convert it to degrees first: 1 radian ≈ 57.2958°. For example, 3.5 radians → 3.5 × 57.2958 ≈ 200.53°
Add 360° to find a positive coterminal angle: 200.53 + 360 ≈ 560.53°
Subtract 360° to find a negative coterminal angle: 200.53 - 360 ≈ -159.47°

Example:

Find a coterminal angle for θ = π/4 radians.

Solution:

Since the angle is in radians, we add or subtract multiples of 2π to find coterminal angles:

Negative coterminal angle: π/4 - 2π = -7π/4

Therefore, −7π/4 is coterminal with π/4.

How the Coterminal Angle Calculator Works

Input:

Select whether you want to “Find coterminal angles” or “Check if two angles are coterminal” from the dropdown.
Choose the unit: degrees or multiples of π radians.
Enter the angle to compute coterminal angles, or enter two angles to check their coterminality.
Click the Calculate button.

Output:

Displays positive coterminal angles
Displays negative coterminal angles
Indicates whether two entered angles are coterminal

References

Wikipedia: Coterminal - Etymology and Definitions
Varsity Tutors: Coterminal Angles - Standard Position & Negative Angles

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