Coterminal Angle Calculator

What are coterminal angles?

An angle in standard position starts on the positive x-axis, which is its initial side. It then turns around the origin and stops at its terminal side. A counterclockwise turn gives a positive angle and a clockwise turn a negative one. Two angles are coterminal when they stop on the same terminal side, even if one of them went around more times to get there.

Take 30°, 390° and −330°. The first turns a little way counterclockwise. The second makes one full turn and then the same 30°. The third turns 330° the other way. All three stop on the same ray, so they are coterminal.

The name comes from "co", meaning together, and "terminal", meaning end: angles that end together. Every angle has infinitely many coterminal angles, because you can always add one more turn.

The coterminal angle formula

A full turn is 360°, or 2π in radians, and adding or taking away whole turns never moves the terminal side. A positive n gives a larger angle, a negative n a smaller one, and n = 0 gives back the angle you started with.

Some textbooks write θ ± 360°n and let n be a positive whole number. It is the same set of angles, with the sign moved in front of the 360°.

How to find coterminal angles in degrees

1. For a positive coterminal angle, add 360°.
2. For a negative coterminal angle, subtract 360°.
3. Repeat as often as you like. Each extra turn gives another coterminal angle.

For 120°, that gives 120° + 360° = 480° and 120° − 360° = −240°. Two turns either way give 840° and −600°.

Show the coterminal angles of 120°

When the angle is already large, one subtraction may not be enough to make it negative. 750° − 360° = 390° is still positive. Subtract again to get 30°, and once more to get −330°, the first negative coterminal angle of 750°.

How to find coterminal angles in radians

In radians a full turn is 2π, so you add or subtract 2π instead of 360°. When the angle is a fraction of π, first write 2π with the same denominator, then add or subtract the numerators. For π/4:

1. Write one turn in quarters: 2π = 8π/4.
2. Add it for a positive coterminal angle: π/4 + 8π/4 = 9π/4.
3. Subtract it for a negative one: π/4 − 8π/4 = −7π/4.

The same method turns −4π/3 into an angle between 0 and 2π. One turn is 6π/3, and −4π/3 + 6π/3 = 2π/3.

Show −4π/3 in the calculator

A radian measure without π, such as 2.5 or 10, works the same way. Because 2π is about 6.2832, the answer comes out as a decimal: 10 − 2π ≈ 3.7168. The calculator shows the exact form, 10 − 2π, next to the decimal, so you can copy whichever your teacher wants.

Finding the coterminal angle between 0° and 360°

Most homework asks for the one coterminal angle from 0° up to, but not including, 360°. For a small angle, add or subtract 360° until it lands in that range. For a big angle, counting the turns first is quicker:

1. Divide the angle by 360.
2. Round the result down to a whole number. Down means toward minus infinity, so −1.39 becomes −2.
3. Take away that many turns: θ − 360° × (the rounded number).

For 1252°: 1252 ÷ 360 ≈ 3.48, which rounds down to 3, and 1252° − 1,080° = 172°. For −501°: −501 ÷ 360 ≈ −1.39, which rounds down to −2. Taking away −2 turns means adding two, so −501° + 720° = 219°.

Show the steps for 1252°

Radians work the same way, dividing by 2π. For 25π/4: 25π/4 ÷ 2π = 25/8 = 3.125, so take away 3 turns, which is 6π or 24π/4. That leaves 25π/4 − 24π/4 = π/4.

Here are the angles that come up most often in homework, worked out the same way:

Other ranges: −180° to 180° and your own

Some problems want −180° to 180° instead (−π to π in radians), so that a turn one way is positive and a turn the other way is negative. The atan2 function in most programming languages returns its angle in this range. To get there, find the angle from 0° to 360° first. If it is more than 180°, take away one more turn, so 219° becomes 219° − 360° = −141°.

Show −501° between −180° and 180°

Worksheets also ask for the coterminal angle between −360° and 0°, or for every coterminal angle in a range such as −720° to 720°. Choose Custom in the calculator and type both ends. It lists every coterminal angle in the range, with both ends included.

Find the angle between −360° and 0° for 120°

Smallest positive and largest negative coterminal angle

The smallest positive coterminal angle is usually the same as the angle between 0° and 360°. The exception is an angle that lands exactly on 0°. Zero is not positive, so for 0°, 360° or 720° the smallest positive coterminal angle is 360°.

The largest negative coterminal angle is one turn below the smallest positive one. For −501° that is 219° − 360° = −141°. The two together answer the common worksheet task "give one positive and one negative coterminal angle", and the calculator shows both every time.

Reference angles and quadrants

The reference angle is the acute angle between the terminal side and the x-axis, so it is always from 0° to 90°. Coterminal angles share a terminal side, which means they also share a quadrant and a reference angle. Find both from the angle between 0° and 360°:

−501° lands on 219°, which is between 180° and 270°, so it is in quadrant III with a reference angle of 219° − 180° = 39°. Angles on an axis, such as 0°, 90°, 180° and 270°, are in no quadrant. The calculator gives them the angle to the x-axis instead: 0° for 0° and 180°, 90° for 90° and 270°.

Reference angles let you work out the sine and cosine of any angle from first-quadrant values. For example, sin 219° = −sin 39°. The value is negative because sine is negative in quadrant III.

Common mistakes

• Adding 360 to an angle in radians. π/3 + 360 is not coterminal with π/3. One turn in radians is 2π, so the next angle is 7π/3.
• Adding fractions without a common denominator. π/4 + 2π is 9π/4, not 3π/5. Write 2π as 8π/4 first.
• Thinking θ and −θ are coterminal. 45° and −45° are mirror images across the x-axis, and −45° is coterminal with 315°.
• Giving 360° as the answer between 0° and 360°. The range stops just before 360°, so 720° lands on 0°.
• Measuring the reference angle to the y-axis. It is always measured to the x-axis.

Where coterminal angles are used

Sine, cosine and tangent repeat every full turn, so you can swap any angle for its coterminal angle between 0° and 360° before you evaluate it: sin 750° = sin 30° = 1/2. Rotation works the same way. A wheel that has turned 1,080° has made three full turns and points the way it started. Game engines and robot controllers keep headings in a fixed range for that reason.

Compass bearings also run from 0° to 360°, though they are measured clockwise from north rather than counterclockwise from the positive x-axis. Turning a running total of heading changes into a bearing is the same as finding the coterminal angle between 0° and 360°.