Coterminal Angle Calculator

θ′ = θ + 360°n coterminal = θ − 360°·⌊θ ÷ 360°⌋

θ

The angle you type, in degrees. It can be negative and it can be far larger than a full turn; both are ordinary inputs here rather than special cases

n

Any whole number, positive, negative or zero. Each value of n picks out one member of the family of angles that share the terminal side, so there are infinitely many of them and the page prints the two most useful

360°

One full turn. Subtracting it leaves the ray pointing exactly where it was, which is what makes two angles coterminal rather than different

⌊θ ÷ 360°⌋

How many whole turns are taken off, rounded down towards minus infinity — not towards zero. Rounding towards zero would leave a negative angle sitting outside the interval instead of wrapped into it, which is the one place this page can go quietly wrong

Coterminal angle

The representative from 0 up to but not including 360 degrees. This is the figure the trigonometry tables and the unit circle are written against, and it is the one most answers are wanted in

Negative coterminal angle

The same ray described as a negative rotation: the reduced angle minus one whole turn, landing from minus 360 up to but not including zero. It is always exactly 360 below the other figure

Two decimal places

How wide the readings are printed. Angles are usually measured to a fraction of a degree rather than to a fraction of a second, and two decimals is about as fine as a protractor reading can honestly be

Whenever an angle has been wound past a full turn and an answer inside one turn is wanted. Shafts and rotating machinery are the plainest case: a coupling turned through three and a quarter revolutions has a final orientation that is stated as an angle under 360, and the same reduction applies to a cam, a crank, a turntable or a robot joint whose travel is quoted in revolutions. Winding and coiling is the second: a coil wound on 14 times, a cable paid out and reeled back, a spring turned through a known number of turns — all of these are counted in revolutions and needed in degrees. Measurement instruments are the third: a phase angle accumulated over many cycles of a signal, a total angle read off an encoder that counts up without resetting, an accumulated bearing error, and any instrument that reports an angle that has been allowed to run past one turn. Navigation and surveying use it for bearings that have been added and subtracted until they leave the 0 to 360 range, and for the difference between two headings, which is why a heading of minus 30 degrees is routinely rewritten as 330. In mathematics it is the first step in almost every trigonometric calculation — a table of sine and cosine values only covers one turn, so any angle outside that range has to be brought back into it before the table can be used. In the classroom it is the visual demonstration of periodicity: draw the same terminal side for 30, 390 and 750 degrees, notice that all three arms land on the same line, and the reason sine repeats every 360 degrees stops being a rule to memorise. And in everyday arithmetic it is the handful of cases where an angle has been subtracted from another and come out negative, which is the point at which most people first want a 0 to 360 answer rather than a signed one.

1. An angle of 400 degrees

   1. Count the whole turns: 400 ÷ 360 = 1.11, so one turn comes off
   2. Subtract it: 400 − 360 = 40
   3. Subtract another turn for the negative representative: 40 − 360 = −320

   The pair the page loads with, chosen because it is only just past a full turn — a reader who has never met the idea can see the whole operation in one step. The two answers are 360 apart, which is true on every row of the table and is the quickest way to check that a result makes sense.

2. An angle of 750 degrees

   1. Count the whole turns: 750 ÷ 360 = 2.08, so two turns come off
   2. Subtract them: 750 − 720 = 30
   3. Subtract another turn: 30 − 360 = −330

   The same terminal side as an angle of 30 degrees, two whole turns further round, and the pair of rows that shows what the table is for. Reading the reduction off the turns column — two turns off 750 leaves 30 — is exactly how the arithmetic is done by hand, and it is why that column is on the table at all.

3. An angle of minus 30 degrees

   1. Count the whole turns, rounding down: −30 ÷ 360 = −0.08, which rounds down to −1
   2. Add that turn back on: −30 − 360 × (−1) = 330
   3. Subtract a turn for the negative side: 330 − 360 = −30

   The direction most people arrive from, since a negative angle is what you get when one heading is subtracted from another. Note that the negative representative is the original angle, unchanged: minus 30 is already inside the negative interval. The rounding has to go downwards here — rounding towards zero would leave minus 30 as it was, outside the positive interval where it belongs.

4. An angle of 1000 degrees

   1. Count the whole turns: 1000 ÷ 360 = 2.78, so two turns come off
   2. Subtract them: 1000 − 720 = 280
   3. Subtract another turn: 280 − 360 = −80

   Two turns and a remainder of 280, which is where the arithmetic stops being obvious: the answer is close to a full turn rather than close to zero, and a reader who expects the small remainder of the earlier examples will be surprised. The page also accepts a million degrees, which comes out at the same 280 and minus 80 — the number of turns changes, the terminal side does not.

5. An angle of 0 degrees

   1. No whole turns come off: 0 is already inside the interval
   2. The positive representative is 0
   3. Subtract a turn for the negative side: 0 − 360 = −360

   The case that fixes the rule for the negative side. The alternative convention would print zero for both figures, which reads as a page that failed to do anything. Taking the negative representative to be a whole turn below the reduced angle guarantees the two printed figures are never the same, and it holds at this boundary as well as everywhere else.

This page reduces an angle and nothing else: it does not say which quadrant the terminal side falls in, it does not give a reference angle, and it does not evaluate any trigonometric function. Each of those belongs to a page of its own. Only degrees are accepted — there is no radians output, and an angle given in radians has to be converted before it is entered. The input is limited to a million degrees in either direction, which is about two thousand seven hundred and seventy-eight full turns; beyond that the figure stops corresponding to anything anyone measures, though the arithmetic would keep working. The two figures printed always differ by exactly 360 degrees, and the negative one is always the lower of the two, so a negative input can produce a positive output and the other way round. Strictly, coterminal angles are defined as non-coincident angles whose terminal sides coincide, so the angle you enter is not coterminal with itself in that strict sense — it coincides with itself. The page follows the everyday usage in which any angle on the same terminal side counts, and says so here rather than in a footnote. Angles are rounded to two decimal places, which is a display width rather than a claim of precision. Nothing here handles degrees, minutes and seconds as separate units, and nothing handles angles measured in gradians or turns.

I entered 40 and the coterminal angle came back as 40. Did anything happen?

Nothing needed to: 40 is already inside a single turn, so it is its own representative and the reduction leaves it alone. Strictly, coterminal angles are defined as angles that are not coincident but finish on the same ray, so an angle is not coterminal with itself under that definition — it coincides with itself. The page follows the everyday usage in which anything on the same terminal side counts, so the first figure repeats your input whenever your input is already reduced. The second figure is the one that changes.

Why is the negative angle a whole turn below rather than just the same angle written negative?

Because the obvious alternative breaks at zero. If the negative representative were taken to be anything from minus 360 up to zero, then an input of zero would print zero and zero — two identical figures, which reads as a page that did nothing. Taking it to be exactly one turn below the reduced angle means the two printed figures are never equal, and they always differ by exactly 360 degrees, which you can check on every row of the table.

Can I enter a negative angle?

Yes, and it is one of the commonest inputs. Negative angles turn up whenever one heading is subtracted from another or a rotation is measured anticlockwise, and minus 30 degrees is routinely rewritten as 330. The reduction rounds downwards rather than towards zero for exactly this reason: rounding towards zero would leave minus 30 sitting outside the interval instead of wrapping it in.

Why are the two answers always 360 degrees apart?

Because the second is the first minus one whole turn, by construction. Subtracting 360 degrees from an angle leaves the ray pointing exactly where it was, so the two figures describe the same terminal side and differ by exactly one revolution. It is the quickest sanity check on any result this page gives you: add 360 to the lower figure and you should land on the upper one.

Does this tell me which quadrant the angle is in?

No — it reduces the angle and stops there. The quadrant a terminal side falls in, and the reference angle between that side and the nearest axis, are the subject of the reference angle and unit circle pages. The reduction this page does is the step that comes before either of them, which is why it is a page of its own rather than a line in someone else's output.

Can I enter radians?

Not on this page — it takes degrees only, and there is no units switch. An angle in radians has to be converted first, by multiplying by 180 and dividing by π. The reduction itself is the same idea in either unit, but the page would have to know which one you meant before it could decide what a full turn is.

• Coterminal Angle — angles whose terminal sides coincide, with the definition this page implements and the n = 0 case that is coincident rather than coterminal — Wolfram MathWorld (United States)
• Angle — what an angle is, including angle standard position, from which the initial side and the terminal side are defined — Wolfram MathWorld (United States)
• Trigonometry — why an angle is reduced into a single turn before a table of values can be used, which is the commonest reason to want this reduction — Wolfram MathWorld (United States)
• 教育部关于印发义务教育课程方案和课程标准（2022 年版）的通知——The fifth item in the annex list of this notice is the Mathematics Curriculum Standards for Compulsory Education (2022 edition); angles and general angles are part of the mathematics curriculum at the compulsory-education and upper-secondary levels, and the wording of the standards and the grade-band breakdown are governed by that annex — 中华人民共和国教育部