Coterminal Angle Calculator

What this calculator does

So, you want angles that point the same way as a given one, or you want to check whether two angles really do point the same way. This tool does both. In one mode it finds angles coterminal with yours; in the other it tells you whether two angles you enter are coterminal.

You pick the mode from a dropdown, choose whether you are working in degrees or in multiples of pi, and enter your angle or angles.

How to use it

1. Choose find a coterminal angle, or check if two angles are coterminal.
2. Pick the unit: degrees or pi radians.
3. Enter your angle, or both angles for the check, and press Calculate.

What coterminal angles are

Two angles are coterminal when they finish in the same place. Drawn from the positive x-axis, they land on the same terminal side, even though the numbers are different. This happens because you can spin all the way around, a full 360 degrees, and end up exactly where you started. So 60 degrees and 420 degrees point in the same direction: 420 is just 60 with one extra full turn added on. Every angle has endlessly many coterminal partners, each one a whole number of full turns away.

Finding coterminal angles

In find mode, the tool first reduces your angle to the standard range, between 0 and a full turn, by removing whole turns. Then it lists coterminal angles in both directions: several positive ones, made by adding one, two, three, and four full turns, and several negative ones, made by subtracting them. In degrees a full turn is 360, so it steps by 360 each time; in pi radians a full turn is 2, so it steps by 2. The list makes plain that coterminal angles march off in both directions forever, spaced exactly one full turn apart.

Checking whether two angles are coterminal

In check mode, the tool takes your two angles and looks at their difference. If that difference is a whole number of full turns, the angles are coterminal; if it is not, they are not. In degrees it tests whether the difference divides evenly by 360, and in pi radians whether it divides evenly by 2. This is the clean test behind the idea: two angles share a terminal side precisely when they differ by a whole number of complete rotations, nothing more and nothing less.

A worked example

In find mode, enter 60 degrees. The tool reports positive coterminal angles of 420, 780, 1140, and 1500 degrees, and negative ones of minus 300, minus 660, and so on, each a full turn from the last. In check mode, enter 60 and 420: their difference is 360, a whole turn, so the tool says yes, they are coterminal. Enter 60 and 400 instead: the difference is 340, not a whole turn, so it says no.

Why coterminal angles matter

Coterminal angles are useful because pointing the same way means behaving the same way. Any two coterminal angles have identical sine, cosine, and tangent values, since those depend only on the terminal side. So when an angle comes out awkwardly large or negative, you can swap it for its neat coterminal partner between 0 and a full turn without changing any of its trig values. That is often the first step in tidying an angle up. If instead you want the acute angle an angle makes with the x-axis, use the reference angle calculator.

Questions people ask

What makes two angles coterminal?

They end on the same terminal side when drawn from the positive x-axis. That happens exactly when they differ by a whole number of full turns.

How many coterminal angles does an angle have?

Infinitely many, in both directions, each one a whole number of full turns, 360 degrees or 2 pi, away from the others.

How does the check work?

It subtracts one angle from the other and sees whether the result is a whole number of full turns: a multiple of 360 degrees, or of 2 in pi radians.

Do coterminal angles have the same trig values?

Yes. Because they share a terminal side, their sine, cosine, and tangent are identical.

How is this different from a reference angle?

A coterminal angle ends on the same side as the original. A reference angle is the acute angle between the terminal side and the horizontal axis.

References

On coterminal angles. Coterminal angles share the same terminal side and differ by a whole number of full rotations, so they carry identical trigonometric values.

1. "Coterminal and Reference Angles," Virtual Math Learning Center, Texas A&M University, on angles in standard position that share a terminal side.
2. "Angles," Precalculus 2e, OpenStax, on finding coterminal angles by adding or subtracting full rotations of 360 degrees or 2 pi.