What are coterminal angles?

Coterminal angles are angles that share the same terminal side when drawn in standard position on a coordinate plane. You get them by adding or subtracting any multiple of 360 degrees from the original angle. For example, 30 degrees, 390 degrees and minus 330 degrees are all coterminal because they all end at the same position on the unit circle.

How do you find the principal coterminal angle?

The principal coterminal angle is the equivalent angle between 0 degrees and 360 degrees (exclusive). To find it, divide the angle by 360, take the remainder (modulo 360) and adjust for negatives. For example, 390 mod 360 = 30 degrees. For minus 30 degrees, add 360 to get 330 degrees.

Can coterminal angles be negative?

Yes. A negative coterminal angle is found by subtracting 360 degrees (or a multiple of 360) from the principal angle. For 30 degrees, subtracting 360 gives minus 330 degrees, which is a valid negative coterminal. Both angles end at the same position on the unit circle.

Coterminal Angle Calculator

Coterminal angles are angles that may look different on paper but point in exactly the same direction on the unit circle. When you draw an angle in standard position with the vertex at the origin and the initial side on the positive x-axis, two angles are coterminal if their terminal sides land on exactly the same ray. You get coterminal angles by adding or subtracting any whole multiple of 360 degrees from the starting angle. The angle 30 degrees, for instance, is coterminal with 390 degrees, 750 degrees, minus 330 degrees and minus 690 degrees, among infinitely many others. In trigonometry this matters because sine, cosine, tangent and the other trig functions produce the same output for any coterminal angle. So if you are asked for sin(390 degrees) you can simplify to sin(30 degrees) = 0.5 without any additional work. The principal coterminal angle is the unique equivalent between 0 and 360 degrees (not including 360). This calculator takes any angle you enter in degrees, including large angles greater than 360, angles greater than 720, and negative angles, and returns the principal coterminal angle, two positive coterminals, two negative coterminals, and the radian equivalent of the principal angle. You can use it for any subject where angle reduction is needed: trigonometry, physics, engineering, or navigation.

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Angle (degrees)

deg

30°

principal coterminal angle (0 to 360)

In radians0.5236 rad

Positive: +360390°

Positive: +720750°

Negative: -360-330°

Negative: -720-690°

Coterminal angles share the same terminal side. Adding or subtracting any multiple of 360 degrees produces a coterminal angle. All share the same sine, cosine and tangent values.

How it works

The principal coterminal angle is found by computing principal = ((angle mod 360) + 360) mod 360. The modulo operation finds the remainder when dividing by 360, and the double application handles negative inputs correctly. Positive coterminals are principal + 360 and principal + 720. Negative coterminals are principal minus 360 and principal minus 720. The radian equivalent is principal × π / 180.

Worked example

Starting angle = 390 degrees. 390 divided by 360 = 1 remainder 30, so the principal coterminal angle is 30 degrees. In radians: 30 × π / 180 = 0.5236 rad. Positive coterminals: 30 + 360 = 390 degrees; 30 + 720 = 750 degrees. Negative coterminals: 30 minus 360 = minus 330 degrees; 30 minus 720 = minus 690 degrees. All of these angles end at the same terminal side, so sin(390) = sin(30) = 0.5.

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