How this instrument works
An angle in standard position starts along the positive x-axis and sweeps around to a terminal side; two angles are coterminal when that sweep ends at the identical ray, no matter how many extra laps one of them took to get there. Because one full lap is 360°, coterminal angles are exactly those separated by a whole-number multiple of 360° — 40°, 400°, 760°, and −320° all terminate at the same direction, 40° past the positive x-axis.
The formula coterminal = θ mod 360° works because a modulo operation is defined to hand back the remainder once every whole multiple of the divisor has been removed, and removing a whole 360° turn is precisely the move that leaves a rotating ray untouched. Feed in 400° and mod 360° divides out one complete turn, returning the 40° left over; feed in a negative angle like −30° and the same operation adds a turn instead of subtracting one, landing on 330°, because the result is always pinned to the half-open interval [0°, 360°).
A genuinely odd consequence follows: every angle has infinitely many coterminal partners stretching in both directions — θ + 360°, θ + 720°, θ − 360°, and on forever — yet only one of them sits inside a single turn. That is also why the reduction is so total: an angle like 36,000°, a hundred complete turns, still collapses to 0°, and no amount of extra spinning changes which way something ends up facing.
- Type any angle, positive, negative, or larger than 360°, into the Angle (degrees) field — 400, −750, and 10,800 are all valid.
- Read the result in Coterminal angle in [0°, 360°) — it is always the one direction that matches your input inside a single turn.
- To check the identity yourself, subtract that result from your original entry; the difference will be an exact whole-number multiple of 360°.
- Try a negative entry and watch the sheet add back whole turns, rather than subtract them, until the answer lands inside [0°, 360°).
Worked example — a 400° sensor reading
A robotic arm's rotation sensor reports a raw reading of 400° after its calibration sweep overshoots one full circle. Enter 400 into the Angle (degrees) field, and Coterminal angle in [0°, 360°) returns 40.0 — the arm's true bearing, since 400° is exactly 360° plus 40° more, one complete revolution and a bit.
Subtracting confirms it cleanly: 400 − 40 = 360, one full turn with nothing left over. Add that turn back the other way and 40 + 360 = 400 again, so the two readings are provably interchangeable — same ray, same sine and cosine, same direction any protractor on the arm would measure.
Questions
What does 'coterminal' actually mean?
Two angles are coterminal when they share the same initial side and the same terminal side — the same final direction — even though the amount of rotation differs by a whole number of 360° turns. 40°, 400°, and 760° are coterminal because each equals 40° plus some number of complete laps.
Why does taking θ mod 360° find the coterminal angle?
Mod 360° returns the remainder left over once every complete 360° turn is divided out, and a complete turn returns a ray to its starting position. The remainder is exactly the leftover rotation that determines where the terminal side ends up — everything before the last full lap has no effect on direction.
How does this handle a negative angle like −30°?
It adds back whole turns until the result lands inside [0°, 360°): −30° + 360° = 330°, so −30° and 330° are coterminal. A negative angle just means rotating clockwise instead of counterclockwise, and one counterclockwise turn brings it into the same positive range.
Is a coterminal angle the same thing as a reference angle?
No. A reference angle is the acute angle between the terminal side and the x-axis, always between 0° and 90°, while a coterminal angle shares the exact same terminal side as the original and can be anywhere in [0°, 360°) — a related idea, not the same one.
Do sine, cosine, and tangent give the same value for coterminal angles?
Yes, always. Sin, cos, and tan each have a period of 360°, so sin(θ) = sin(θ + 360°k) for any integer k. That periodicity is the entire reason coterminal angles are useful: reducing an unwieldy angle like 4,000° to its 0°–360° equivalent never changes any trig ratio computed from it.
What happens when the angle is an exact multiple of 360°?
It reduces to exactly 0°. 720°, two full turns, and −360°, one full turn backwards, both return to the starting ray with nothing left over, so their coterminal angle is 0.0°.