Coterminal Angle Calculator

What a coterminal angle is

Two angles are coterminal when they end on the same side of the circle. Start on the positive x-axis, rotate, and stop. Any other angle that stops on the same spoke is coterminal with the first one, because the extra rotation only carried you around a whole number of times. In degrees the difference is a multiple of 360. In radians it is a multiple of 2 pi, which is 6.28319.

OpenStax Precalculus, section 5.1 defines an angle's measure as the rotation from the initial side to the terminal side and calls one full circle 360 degrees. Nothing in that definition stops you from rotating past one turn, so 400 degrees is a legal angle, and its terminal side lands exactly where the terminal side of 40 degrees lands.

That gives a family rather than a pair. For 40 degrees the family runs 40, 400, 760, 1120 in one direction and -320, -680, -1040 in the other. The calculator above accepts any member of the family and returns the rest, together with the quadrant, the reference angle and the trig values that every member shares.

Normalizing into the 0 to 360 range

The standard normalization is a modulo: normalized = angle - 360 x floor(angle / 360). For 400 degrees, 400 / 360 = 1.111 and floor gives 1, so 400 - 360 = 40. For -75 degrees, -75 / 360 = -0.208 and floor gives -1, so -75 + 360 = 285. For 1000 degrees, floor(2.777) = 2 and 1000 - 720 = 280.

Every row in that table has the same sine, cosine and tangent as the angle you started with, which is the practical reason the normalization earns its keep. The turns-removed column is the k in the definition: 400 is 40 plus 360 times 1, and -75 is 285 plus 360 times -1.

A sanity check that catches most slips: the answer has to sit in the same place on the circle. If you normalize -40 and land on 320, you subtracted when you needed to add. If you normalize 400 and land on 40 but report the quadrant as the fourth, the quadrant logic walked the wrong way around.

Radians use 2 pi, not 360

Radians change the period, not the idea. One full turn is 2 pi radians, 6.28319, which follows from the NIST SI reference definition of the radian as the angle subtended by an arc whose length equals the radius. To move one turn you add or subtract 6.28319, never 360.

Take 7 pi / 6, which is 210 degrees and 3.66519 rad. Subtract 2 pi and you get -5 pi / 6, which is -2.6180 rad and lands on the same spoke. Add 2 pi instead and you get 19 pi / 6, which is 9.94838 rad or 570 degrees. All three point the same way.

The mistake to avoid is subtracting 360 from a radian angle. 3.66519 - 360 = -356.33 looks like a subtraction that worked, but 360 radians is 57.3 full turns, so the answer has nothing to do with the original direction. Subtract 6.28319 and the result stays in the neighbourhood of the angle you meant.

Read the table as a conversion kit. Degrees and radians measure the same rotation, so a coterminal step of 360 degrees is always a step of 6.2832 radians. Mixing the two inside one calculation is the single most common error on homework sets and in spreadsheet formulas.

Quadrants, axes and the reference angle

The normalized angle tells you which quarter of the circle you are in. Anything from 0 to 90 is the first quadrant, 90 to 180 the second, 180 to 270 the third, 270 to 360 the fourth. Multiples of 90 sit on an axis and belong to no quadrant: 180 degrees lies on the negative x-axis, and its sine is 0.

Do not confuse a coterminal angle with a reference angle. A coterminal angle answers the question "same direction, how many turns away". A reference angle answers "how far is this from the nearest part of the x-axis", and it always falls between 0 and 90. For 210 degrees the coterminal answers are 570, -150 and 930, while the reference angle is 30. Both come out of the same picture, and swapping them changes the answer to a different question.

The reference angle is what makes a trig table usable beyond the first quadrant: sin(210) has the same size as sin(30) and takes the negative sign from the third quadrant, so the value is -0.5. That is a two step reduction, and the coterminal step comes first.

Sine, cosine and tangent stay the same

Because the terminal side does not move, the trig values do not move either. sin 40 = 0.6428 and sin 400 = 0.6428; cos sits at 0.766 for both, tan at 0.8391 for both. Run it the other way and -75 and 285 both give sin -0.9659, cos 0.2588 and tan -3.7321.

Tangent has a shorter period than the other two. NIST DLMF section 4.21 gives the periods as 2 pi for sine and cosine and pi for tangent, so 40 degrees and 220 degrees share tan 0.8391 while their sines carry opposite signs. That is why a coterminal check is a sound way to audit a sine or cosine table entry and a poor way to audit a tangent sign.

The same rule is what makes angle reduction work in code: sin(400) is evaluated as sin(40) because the terminal side is identical, so a controller that wraps a crank angle still returns the right value after a full extra turn.

Finding a coterminal angle in a range you choose

Homework asks for a coterminal angle "between 0 and 360" or "between -360 and 0", and the range changes the answer without changing the direction. 40 degrees becomes -320 in the turn below zero. 285 becomes -75. 1000 becomes -80, because that range needs three turns removed instead of two.

The formula for any range is k = ceil((low - angle) / 360), then answer = angle + 360k. With low at -360 and an angle of 1000, (-360 - 1000) / 360 = -3.778 and ceil gives -3, so 1000 - 1080 = -80. With low at 0, ceil(-2.778) = -2 and 1000 - 720 = 280. Choose the range first, then do one division.

In radians the same move divides by 2 pi. For 7 pi / 6 into the range -2 pi to 0, k = ceil((-6.28319 - 3.66519) / 6.28319) = ceil(-1.583) = -1, so 3.66519 - 6.28319 = -2.6180, the -5 pi / 6 from earlier. The arithmetic is identical; only the size of the turn changes.

Where coterminal angles turn up

Surveying and machine work are the two places where this stops being homework. A total station accumulates turning angles well past 360 degrees, and surveying software normalizes each shot to a bearing between 0 and 360 before it computes coordinates. A rotary table on a mill can be wound 730 degrees; the control reduces that to 10 to find the work angle. Clock hands behave the same way, since 75 minutes is 450 degrees of minute-hand travel and 90 degrees of position.

Trig tables are the third place. sin(400) is not a separate entry, it reduces to sin(40), and every printed table relies on that reduction to stay one page long. Programs that store headings wrap them the same way, because a heading of 370 and a heading of 10 name one direction, and the geometry downstream needs a single convention.

One last check before you trust a number: the difference between the input and the answer should divide by 360 with no remainder. 1000 minus 280 is 720, which is two turns exactly. If the difference is 700, an arithmetic slip crept in somewhere, and no amount of quadrant reasoning will repair it.