About the Coterminal Angle Calculator
Two angles are coterminal when they start on the positive x-axis and end on the same terminal side, which happens whenever they differ by a whole number of full turns (360° or 2π radians). This coterminal angle calculator takes any angle — positive, negative, larger than a full circle or in radians — and returns its principal angle in the range 0° to 360°, the smallest positive and largest negative coterminal angles, and a list of coterminal angles on either side.
It is aimed at trigonometry and precalculus students who need to reduce large angles before using the unit circle, and at anyone checking homework where an answer like 1,110° has to be written as 30°. Because coterminal angles share a terminal side, they also share every trigonometric value: sin 400° equals sin 40°.
Enter the angle, choose degrees or radians, and optionally pick a number of turns k to add or subtract. Results are given in the same unit you entered; radian answers are also shown as multiples of π.
With the default inputs, the principal angle (0 to one full turn) is 40. Change any value above to recalculate instantly.
How to use the coterminal angle calculator
- 1Enter the angle — it can be negative or larger than a full turn.
- 2Choose whether it is in degrees or radians.
- 3Read the principal angle and the nearest positive and negative coterminal angles.
- 4Set k to jump a specific number of full turns forward or backward.
- 5Use the table for a list of coterminal angles on both sides.
Formula and method
An angle and any angle that differs from it by a whole number of full rotations end in the same place, so every coterminal angle is θ + 360°·k in degrees or θ + 2π·k in radians, where k is any integer. The principal angle is the one in the interval [0°, 360°), found with a positive modulo: subtract 360° until the angle is below 360°, or add 360° until it is non-negative.
The smallest positive coterminal angle is the principal angle itself (or one full turn if the principal angle is 0), and the largest negative coterminal angle is the principal angle minus one full turn. The calculator rounds tiny floating-point noise so 720° reduces cleanly to 0°.
- θ
- The angle you entered
- k
- Any integer number of full turns
- θc
- A coterminal angle
Worked examples
Coterminal angles of 400°
400° − 360° = 40°, so the principal (and smallest positive) coterminal angle is 40°. Subtracting another 360° gives −320°, and adding two full turns gives 400° + 720° = 1,120°.
A large negative angle: −1,000°
Adding three full turns gives −1,000° + 1,080° = 80°. The largest negative coterminal angle is 80° − 360° = −280°, and −1,000° + 360° = −640°.
10 radians
One full turn is 2π ≈ 6.2832 rad, so 10 − 6.2832 ≈ 3.7168 rad is the principal angle. Subtracting another 2π gives about −2.5664 rad.
Frequently asked questions
How do you find a coterminal angle?+
Add or subtract 360° (or 2π radians) any number of times. For 50°, coterminal angles include 410°, 770°, −310° and −670°.
How many coterminal angles does an angle have?+
Infinitely many. You can add or subtract any whole number of full turns, so there is one coterminal angle for every integer k.
Do coterminal angles have the same sine and cosine?+
Yes. Coterminal angles share the same terminal side, so all six trigonometric functions give identical values — sin 400° = sin 40° ≈ 0.6428.
How do I find coterminal angles in radians?+
Add or subtract 2π. For π/3, coterminal angles are 7π/3, 13π/3, −5π/3 and so on. In decimals, 2π ≈ 6.2832.
What is the difference between a coterminal angle and a reference angle?+
A coterminal angle ends on the same terminal side; a reference angle is the acute angle between the terminal side and the x-axis. 150° has coterminal angle 510° and reference angle 30°.