Coterminal Angle Calculator

Why there are infinitely many

Turning through 360° brings you back to where you started. So does turning 720°, or −360°, or any whole number of turns.

That means every direction is described by infinitely many angles. The ray at 70° is also at 430°, 790°, 1150°, −290°, −650°, and so on without end — the set 70° + 360n for every integer n.

A question asking for “a coterminal angle” wants one member. A question asking for “all coterminal angles” wants that expression. This page gives both, because which one is correct depends entirely on how the question was worded.

The principal angle

Of the infinitely many, one is picked as the standard representative: the one between 0° and 360°. It is called the principal angle.

Finding it is a single division. Divide by 360, take the whole number of turns, and subtract that many turns. For 1000°: two whole turns, so 1000 − 720 = 280°.

For negatives the division rounds DOWN rather than towards zero, which is what keeps the answer positive. −100° is “minus one turn plus 260°”, so the principal angle is 260°, not −100°. This is the same choice the floored convention makes for the modulo operator.

Why the trig values match

Sine, cosine and tangent depend only on where the ray finishes. They know nothing about how many times it went round to get there.

So sin 430° = sin 70° exactly, and cos(−290°) = cos 70° exactly. This is the periodicity of the trigonometric functions, and it is why they are periodic: coterminal angles are literally the same direction.

It is also why reducing first is safe. Any angle can be brought into one turn before evaluating anything, with no loss and no approximation.

Radians, if you need them

In radians a full turn is 2π, so coterminal angles differ by 2πn rather than 360n.

The idea is identical; only the unit changes. 7π/3 is coterminal with π/3, because they differ by 2π. And −π/2 is coterminal with 3π/2.

This page works in degrees because that is what the question is usually asked in. To convert, multiply degrees by π/180 — and note that the reduction is exact in degrees and only approximate in radians, since π is irrational and any decimal form of it is not.

Where it matters

Evaluating trig functions. Reduce first, then work in one turn. It is the step before finding a reference angle.

Rotations in code. A rotation that accumulates will drift past 360° and keep going. Normalising it into one turn keeps the number small and the comparisons meaningful.

Bearings and headings. A compass bearing is by convention between 0° and 360°, so anything outside gets reduced. Turning 45° left from a heading of 20° gives −25°, which is reported as 335°.

Anything periodic. Phase in a wave, position on a dial, time on a clock face: all of them are angles modulo one turn.

Reducing by division, not by looping

The obvious way to reduce an angle is to keep subtracting 360 until it is in range. It works, and it is a bad way to do it.

For 36,000,000.5° that is a hundred thousand subtractions, and in floating point every one of them loses a little precision, so the half degree at the end is no longer exactly a half.

This page divides once, on exact fractions. The answer to that input is 0.5°, exactly, and it arrives immediately. The same approach keeps a fractional input like a third of a degree intact rather than turning it into 0.3333333333.

Sources and methodology

The convention is standard; these are the references for it and for the arithmetic underneath.

• National Council of Teachers of Mathematics — where coterminal angles and the principal value are defined
• NIST SP 811 — plane angle conventions in SI
• Modulo Calculator — the same reduction, on numbers rather than angles
• How we calculate — methodology

Method. The reduction is one exact division on rationals: subtract 360 times the floor of θ/360, computed with integer arithmetic rather than by adding 360 in a loop. That is what lets a thirty-six-million-degree angle reduce instantly and exactly, and it keeps a fractional part like 0.5° intact. The suite asserts that the principal angle always lands in [0°, 360°) across several thousand generated inputs on both sides of zero. That engine is verified on every change against 82 hand-written assertions, including that the principal angle always lands in [0°, 360°) across a thousand angles spanning ten full turns in each direction. The count and the per-case breakdown are published on the formula verification page.

Related calculators

Where this goes next:

More in Math, or browse all calculators.

Read the guide

Once reduced, the Reference Angle Calculator takes the next step — the acute angle to the x-axis, and the sign each trigonometric function carries there.