Coterminal Angle Calculator
Find coterminal angles in degrees or radians: the principal angle in [0°, 360°), the largest negative coterminal angle, and a list of positive and negative coterminal angles.
How it works
Coterminal angles: θ + 360°·k (degrees) or θ + 2π·k (radians), k = ±1, ±2, … · principal angle = θ mod 360°
Two angles in standard position are coterminal when they share the same terminal side, so they differ by a whole number of full turns (360° or 2π radians). Adding or subtracting 360° any number of times gives every coterminal angle; reducing θ modulo 360° gives the principal angle between 0° and 360°, the one used to read off trig values on the unit circle.
Worked example
For θ = 420°: 420° − 360° = 60°, so the principal coterminal angle is 60° (one full turn removed). The largest negative coterminal angle is 60° − 360° = −300°, and further coterminal angles are 780°, 1,140° and 1,500° (adding turns) or 60°, −300° and −660° (subtracting turns).
Sources: Coterminal angle — Wolfram MathWorld · Angles in standard position — OpenStax Precalculus 7.1
Frequently asked questions
How do you find a coterminal angle?
Add or subtract 360° (2π in radians) as many times as you like. For a positive coterminal angle of −45°, add 360° to get 315°; for a negative one of 100°, subtract 360° to get −260°.
How many coterminal angles does an angle have?
Infinitely many — one for every whole number k in θ + 360°·k. Textbooks usually ask for the one between 0° and 360° (or 0 and 2π) plus one positive and one negative example.
How do I find coterminal angles in radians?
Use 2π instead of 360°. For 7π/4, subtracting 2π gives −π/4 and adding 2π gives 15π/4. Choose “Multiples of π” and enter 1.75 for 7π/4.
Do coterminal angles have the same sine and cosine?
Yes. They end on the same ray, so every trigonometric function (sin, cos, tan and their reciprocals) has the same value for all coterminal angles.
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This calculator is for educational purposes. Double-check important results before acting on them.