Coterminal Angle Calculator | Formula

This coterminal angle calculator converts an angle in degrees or radians into four coterminal angles: two found by adding 1 and 2 full rotations, and two found by subtracting 1 and 2 full rotations.

Coterminal Angle Calculator

Coterminal Normalize Reference & Quadrant Select Radians or Degrees for the Angle Degrees Radians Common Angles (optional) Angle (Degrees or Radians) Radians or Degrees Degrees Radians Coterminal Angle (+1 full rotation) Coterminal Angle (+2 full rotations) Coterminal Angle (−1 full rotation) Coterminal Angle (−2 full rotations) Select Radians or Degrees for the Angle Degrees Radians Common Angles (optional) Angle (Degrees or Radians) Normalization Range [0, 360) or [0, 2π) (-180, 180] or (-π, π] Radians or Degrees Degrees Radians Normalized Angle Select Radians or Degrees for the Angle Degrees Radians Common Angles (optional) Angle (Degrees or Radians) Radians or Degrees (for Reference Angle and Coterminal Output) Degrees Radians Reference Angle Quadrant / Axis Non-negative Coterminal Angle Calculate Reset

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Coterminal Angle Formula

The following formulas are used to calculate coterminal angles.

Coterminal angles (degrees) = Angle + 360k, k ∈ ℤ Coterminal angles (radians) = Angle + 2πk, k ∈ ℤ

To calculate coterminal angles, add and/or subtract one full rotation from the current angle: 360° in degrees or 2π in radians.

The formulas above show the general rule using an integer k. In reality, there is an infinite number of coterminal angles. The definition is simply an angle that ends at the same point as another angle on a coordinate plane. Since the unit circle has a total rotation of 360 degrees (2π radians), adding or subtracting that amount yields a coterminal angle, as does the coterminal angle calculator above.

To calculate a coterminal angle between 0 and 2π (in radians), add or subtract 2π until the result is in the desired interval. For example:

Find the coterminal angle of 11π/3:

Since 11π/3 > 2π, subtract 2π from the given angle. (If the given angle were negative, you would add 2π until it falls between 0 and 2π.)

11π/3 - 2π = 11π/3 - 6π/3 = 5π/3.

Coterminal Angle Definition

A coterminal angle is an angle that ends at the same geometric point on the coordinate plane as another angle.

Common Coterminal Angle Values

The following list outlines the coterminal angles of the most commonly calculated angles.

How to calculate a coterminal angle

Let's take a look at an example of how you might calculate the coterminal angle.

1. Let's assume you are given an angle in radians. For this example, the angle is 2.5 radians.
2. First, you must convert the radians to degrees. 1 radian is approximately equal to 57.2958 degrees, so 2.5 × 57.2958 ≈ 143.2395 degrees.
3. Last, we need to add 360 degrees to that angle to find an angle that is coterminal with the original angle, so 143.2395 + 360 = 503.2395 degrees.
4. Check the answer using the calculator above.

FAQ

What is a coterminal angle?

A coterminal angle is an angle that ends at the same location as another angle. For example, 1 degree and 361 degrees are at the same location since one full rotation is 360 degrees.

What is the coterminal angle formula?

Coterminal angles differ by a full rotation. In degrees: Angle + 360k. In radians: Angle + 2πk, where k is any integer.