Math
Coterminal Angle Calculator
Coterminal angles share the same terminal side, so they differ by whole turns of 360° (2π radians).
Enter any angle and get the smallest positive coterminal angle, the largest negative one, several more, and the general form θ + 360°k, together with the quadrant and reference angle.
Type a number, a fraction like 180/7, a π expression like 5π/6, or degrees-minutes-seconds like 30° 15′ 30″. Negative and large angles are fine.
Try an example
Result
Smallest positive coterminal angle
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- Largest negative coterminal angle
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- General form
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- More coterminal angles
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- Quadrant
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- Reference angle
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What coterminal angles are
Rotating a full circle brings the terminal side back to the same place, so 30°, 390°, 750°, and -330° all point the same way. Because the terminal side is identical, every trig function has the same value for all of them.
How to find them
- Add or subtract 360° (or 2π) until the angle lands between 0° and 360°. That is the smallest positive coterminal angle (use 360° itself for angles coterminal with 0°).
- Subtract 360° once more to get the largest negative coterminal angle.
- Write the general form by adding 360°k to the reduced angle.
How to read the answer
- The smallest positive coterminal angle is the one between 0° and 360° that you use for The small positive angle between a given angle's arm and the x-axis. and the unit circle.
- The largest negative coterminal angle is the same rotation measured clockwise.
- Both values are shown in degrees and An angle unit where a full turn is 2π instead of 360 degrees., with π fractions when they exist.
Common mistakes and edge cases
- Adding 180° instead of 360°: 180° flips the terminal side to the opposite direction, so 30° and 210° are not coterminal.
- Mixing units: in radians add 2π ≈ 6.2832, not 360.
- Angles like 720° are coterminal with 0°; their smallest positive coterminal angle is reported as 360°.
- Coterminal is not the same as reference: 390° is coterminal with 30° and its reference angle is also 30°, but 150° has reference angle 30° without being coterminal.
Worked examples
Negative angle
-45°
Smallest positive coterminal angle
315° (7π/4)
Large angle
1125° = 3 turns + 45°
Smallest positive coterminal angle
45° (π/4)
Radians
13π/6 reduces to π/6
Smallest positive coterminal angle
π/6 (30°)
Angle coterminal with 0°
720°
Smallest positive coterminal angle
360° (2π)
Negative radians
-π/3
Smallest positive coterminal angle
5π/3 (300°)
Decimal degrees
400.5°
Smallest positive coterminal angle
40.5° (9π/40)
Unreadable input
Smallest positive coterminal angle
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Frequently asked questions
What is a coterminal angle?+
An angle that shares its terminal side with another angle. Coterminal angles differ by a multiple of 360° (2π radians), so 30° and 390° are coterminal.
How do I find a negative coterminal angle?+
Subtract 360° from the angle (or from its positive coterminal angle) until the result is negative. For 30°, the largest negative coterminal angle is 30° − 360° = -330°.
How many coterminal angles does an angle have?+
Infinitely many: one for every integer k in θ + 360°k. This calculator lists the nearest few and gives the general form.
What is the coterminal angle of -45°?+
-45° + 360° = 315°, the smallest positive coterminal angle. -45° is already the largest negative one.
Do coterminal angles have the same sine and cosine?+
Yes. Their terminal sides are identical, so they hit the unit circle at the same point and every trig function agrees.
How do coterminal angles work in radians?+
Add or subtract 2π instead of 360°. 13π/6 − 2π = π/6, so 13π/6 and π/6 are coterminal.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026