Enter any angle in degrees or radians, π forms like −7π/6 included, to get its coterminal angle in [0°, 360°) and [0, 2π), the nearest positive and negative coterminal angles, a full list or every coterminal angle in a range you choose, and a sketch of the terminal side.
Last updated:
Degrees or radians: 45, -210, 1000, 3π/4, -7pi/6, 2.5
Unit
Show
Same unit as the angle (π allowed)
Enter your values to see the result.
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How to use the Coterminal Angle Calculator
Type the angle, for example 45, -210, 1000 or 3π/4.
Choose degrees or radians; anything containing π is read as radians.
Pick what to show: the nearest coterminal angles (set how many) or all coterminal angles in a range.
Press Find coterminal angles; results also update as you type.
Copy the results or download the table of coterminal angles as CSV.
Coterminal angle formulas
Goal
Degrees
Radians
All coterminal angles
θ + 360°·k
θ + 2π·k
Positive coterminal
θ + 360°
θ + 2π
Negative coterminal
θ − 360°
θ − 2π
Principal angle
θ − 360°·⌊θ/360°⌋, in [0°, 360°)
θ − 2π·⌊θ/2π⌋, in [0, 2π)
Are α and β coterminal?
(α − β) ÷ 360° is a whole number
(α − β) ÷ 2π is a whole number
Here k is any integer (…, −2, −1, 0, 1, 2, …). Positive k turns counterclockwise; negative k turns clockwise.
Coterminal angle examples
Angle
θ + 360° (2π)
θ − 360° (2π)
In [0°, 360°)
60°
420°
−300°
60°
−45°
315°
−405°
315°
780°
1140°
420°
60°
−520°
−160°
−880°
200°
π/6
13π/6
−11π/6
π/6 (30°)
5π/4
13π/4
−3π/4
5π/4 (225°)
−2π/3
4π/3
−8π/3
4π/3 (240°)
17π/4
25π/4
9π/4
π/4 (45°)
Tips
Check your answer: two angles are coterminal exactly when their difference is a multiple of 360° (2π).
Big angles: divide by 360 to count the full turns, then keep the remainder. A negative remainder means you add 360° once more.
Radians with π: use a common denominator. 2π = 4π/2 = 6π/3 = 8π/4 = 12π/6.
Same trig values: sin, cos and tan of coterminal angles are equal, so simplify to the principal angle before evaluating.
Frequently asked questions
What are coterminal angles?
Coterminal angles start on the positive x-axis and end on the same terminal side, so they differ only by whole turns. 30°, 390° and −330° are all coterminal because 390° = 30° + 360° and −330° = 30° − 360°.
How do you find coterminal angles?
Add or subtract multiples of 360° in degrees, or multiples of 2π in radians. The general form is θ + 360°·k (or θ + 2πk) where k is any integer. Every angle has infinitely many coterminal angles.
How do I find a positive and a negative coterminal angle?
For a positive one add 360° (or 2π); for a negative one subtract 360°. For 45° that gives 405° and −315°. If the angle is very large or very negative, keep adding or subtracting until you get the sign you want.
How do you find the coterminal angle between 0° and 360°?
Divide the angle by 360°, round down to a whole number of turns, and subtract that many turns: θ − 360° × ⌊θ ÷ 360°⌋. For 1000° that is 1000° − 2 × 360° = 280°; for −210° it is −210° + 360° = 150°.
How do you find coterminal angles in radians?
Add or subtract 2π. Write 2π with the same denominator as the angle first: for 3π/4, 2π = 8π/4, so the coterminal angles are 11π/4, 19π/4, −5π/4, −13π/4 and so on.
Are coterminal angles the same as reference angles?
No. Coterminal angles end on the same terminal side and can be any size. A reference angle is the acute angle between the terminal side and the x-axis, always between 0° and 90°. Coterminal angles always share the same reference angle and the same trig values.
Is an angle coterminal with itself?
Yes. Using k = 0 in θ + 360°·k gives the angle itself, so every angle is trivially coterminal with itself. In practice, questions ask for a different angle, which this calculator lists with k = ±1, ±2 and so on.
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.