Coterminal Angles Calculator
Reduce any angle to its standard position. Enter the angle in degrees or radians and get the coterminal angle in [0°, 360°) — plus the matching negative coterminal and the general 360°k family.
Example: with Angle 420 · Angle unit degrees → Coterminal angle in [0°, 360°): 60°.
- Negative coterminal in [−360°, 0°)-300°
- All coterminal angles60° + 360°k (k = any integer)
Computed by the calculator below using its default values. Change any input to see your own numbers.
Coterminal angles share a terminal side: θ ± 360°k in degrees, θ ± 2πk in radians. 420° and 60° point the same way — one just took an extra lap.
Same direction, different lap count
Stand at the origin and sweep an angle from the positive x-axis. Sweep 60°, or sweep 420°, or back up 300° — your arm ends up pointing the same way every time. Those are coterminal angles: they differ by whole turns of 360° (2π radians), so they share a terminal side and therefore identical values for every trig function. sin(420°) = sin(60°), no exceptions.
That is why the reduction matters. Any angle, however large or negative, collapses to exactly one representative in [0°, 360°), and that representative tells you the quadrant, the reference angle, and the signs of sine and cosine. The calculator computes it as the remainder of the angle divided by 360° (adjusted to be non-negative), then subtracts one turn for the negative twin.
How it’s calculated
Standard coterminal angle = ((θ mod 360) + 360) mod 360 in degrees, or the same with 2π = 6.283185307 in radians — this lands in [0°, 360°) for any input, positive or negative. Negative coterminal = standard − 360° (in [−360°, 0°)). Full family: θ + 360°k (θ + 2πk rad) for every integer k.
Angles are measured in standard position (counterclockwise from the positive x-axis); the tool reports the two representatives nearest zero, not every coterminal angle — there are infinitely many.
Reductions at a glance
| Input angle | Coterminal in [0°, 360°) | Negative coterminal |
|---|---|---|
| 420° | 60° | −300° |
| 780° | 60° | −300° |
| −45° | 315° | −45° |
| 1,000° | 280° | −80° |
| 90° | 90° | −270° |
| 7 rad | 0.7168 rad | −5.5664 rad |
Computed with θ mod 360° (θ mod 2π for radians), shifted into range; 7 − 2π = 0.7168.
Common mistakes
- Adding 180° instead of 360° — 180° apart is the opposite direction (supplement territory), not coterminal.
- Using 360 in radian mode: radians wrap every 2π ≈ 6.2832, so 7 rad reduces to 0.7168 rad, not to 7 − 360.
- Reporting −45° and 45° as coterminal — they differ by 90°, not a full turn. The coterminal partner of −45° is 315°.
- Assuming the reduced angle changes the trig values; the whole point is that sin, cos, and tan are identical for every coterminal angle.
Frequently asked questions
What is the coterminal angle formula?
Coterminal angles are θ + 360°k for any integer k (θ + 2πk in radians). To find the standard one in [0°, 360°), take ((θ mod 360) + 360) mod 360 — that is, keep adding or subtracting 360° until you land between 0° and 360°.
What is the coterminal angle of 420°?
60° is the standard positive one (420 − 360), and −300° is the nearest negative one (60 − 360). Every angle of the form 60° + 360°k qualifies: 780°, 1,140°, −660°, and so on.
Are coterminal angles equal?
Not as rotation amounts — 420° involves more turning than 60°. But they end on the same terminal side, so every trigonometric function gives the same value for both. For trig purposes they are interchangeable.
How do coterminal angles work in radians?
Identically, with 2π playing the role of 360°. For 7 rad: 7 − 2π ≈ 0.7168 rad is the standard coterminal angle, and 0.7168 − 2π ≈ −5.5664 rad is the negative one.
Is 0° coterminal with 360°?
Yes — they differ by exactly one full turn. The calculator reports 0° as the standard representative because the convention uses the half-open interval [0°, 360°), which includes 0 and excludes 360.