Coterminal Angle Calculator

Find angles coterminal with a given angle. Enter an angle in degrees or radians and get the coterminal angle after n rotations, the smallest positive coterminal angle, the reference angle, and the quadrant.

Quick Facts

Formula
Coterminal angle = θ + n × 360° (degrees) or θ + n × 2π (radians)
n is any integer; every full rotation returns to the same terminal side.
Reference angle
The acute angle (0°-90°) between the terminal side and the x-axis
Used to relate any angle's trig values back to a first-quadrant angle.

Your Results

Calculated
Coterminal angle (θ + n rotations)
-
θ plus n full rotations
Smallest positive coterminal angle
-
In the range 0° to 360°
Reference angle
-
Acute angle to the x-axis
Quadrant
-
Where the terminal side falls

Ready

Enter an angle, choose degrees or radians, and set the number of rotations.

Understanding Coterminal Angles

An angle drawn in standard position starts at the positive x-axis (the initial side) and sweeps around to a terminal side. Coterminal angles are any angles that end on that exact same terminal side, even though they represent a different total amount of rotation. Two angles are coterminal whenever they differ by a whole number of complete revolutions.

The formula

For an angle θ, every angle coterminal with it can be written as:

  • Degrees: θ + n × 360°, for any integer n
  • Radians: θ + n × 2π, for any integer n

n can be positive (extra counterclockwise turns), negative (clockwise turns), or zero. Because one full rotation always returns to the same terminal side, adding or subtracting 360° (or 2π radians) never changes where the angle points — only how many times it wound around to get there.

Finding the smallest positive coterminal angle

Most problems ask for the coterminal angle between 0° and 360° (or 0 and 2π radians). To find it, take the angle modulo 360°: if the angle is negative, keep adding 360° until it lands in that range; if it is 360° or larger, keep subtracting 360°. The result — often called the angle's "primary" coterminal angle — is unique for every angle.

Reference angle and quadrant

Once an angle is normalized to the 0°-360° range, its quadrant shows which quarter of the coordinate plane the terminal side falls in (I, II, III, or IV), and its reference angle is the acute angle — always between 0° and 90° — formed between the terminal side and the nearest x-axis. Reference angles matter because sin, cos, and tan of any angle equal ± the same ratio of its reference angle, with only the sign changing by quadrant.

Common uses

  • Simplifying trig expressions for angles outside 0°-360° (or 0-2π) before evaluating sin, cos, or tan
  • Rotational and periodic motion problems in physics and engineering, where an object can wind around many times but only its final position matters
  • Navigation and bearing calculations, where headings are conventionally reported between 0° and 360°

Frequently Asked Questions

What is a coterminal angle?
Two angles are coterminal if they share the same initial side and the same terminal side when drawn in standard position. They differ by a whole number of full rotations: 360° × n in degrees or 2π × n in radians, for any integer n — positive, negative, or zero.
How do you find the smallest positive coterminal angle?
Take the angle modulo 360° (or 2π radians). If the angle is negative, keep adding 360° until the result falls between 0° and 360°; if it is larger than 360°, keep subtracting 360° until it does. The value you land on is unique and is often called the angle's "primary" coterminal angle.
How many coterminal angles does an angle have?
Infinitely many. For any angle θ, every value θ + 360° × n (n = ..., -2, -1, 0, 1, 2, ...) is coterminal with it, since each one completes a whole number of extra turns and ends on the same terminal side.
What is the difference between a coterminal angle and a reference angle?
A coterminal angle shares the same terminal side as the original angle and can be any size, larger or smaller by full rotations. A reference angle is always between 0° and 90° and measures the acute angle between the terminal side and the x-axis — it is used to relate any angle back to a first-quadrant trig value.

Related calculators