Coterminal Angle Calculator
Enter any angle in degrees or radians to find its coterminal angles instantly. The calculator returns the principal coterminal angle in [0, 360) or [0, 2pi), the next three positive and three negative coterminal angles, the reference angle, and the quadrant. A step-by-step panel shows exactly how each value is derived.
Formula
Worked example
For 405 degrees: 405 / 360 = 1.125, so k = 1 full rotation. Principal = 405 - 1 * 360 = 45 degrees (Quadrant I). Reference angle = 45 degrees. Positive coterminals: 765, 1125, 1485 degrees. Negative coterminals: 45, -315, -675 degrees.
What are coterminal angles?
Two angles are coterminal when their terminal sides point in the same direction in standard position, even though the angles themselves differ by one or more full rotations. Standard position means the initial side lies along the positive x-axis and the vertex is at the origin. Because one full rotation equals 360 degrees or 2pi radians, any integer multiple of that period produces an angle with the same terminal side as the original. So 45 degrees, 405 degrees, 765 degrees and -315 degrees all look identical on the unit circle: the ray terminates at the same place no matter how many extra loops were made getting there. This property makes coterminal angles a fundamental concept in trigonometry, because the sine, cosine and tangent of an angle depend only on where its terminal side lands, not on how many full rotations were made to reach that position.
How to find coterminal angles
The formula is straightforward: add or subtract any integer multiple of 360 degrees (or 2pi radians) to produce a coterminal angle. To find the principal coterminal angle, which is the unique angle in [0, 360) or [0, 2pi), use the modulo operation: divide the input by the period, take the floor to find k, then subtract k times the period. For example, 855 degrees divided by 360 gives 2.375, so k = 2 and the principal angle is 855 - 720 = 135 degrees. For negative inputs, the formula still works: -45 degrees gives floor(-45 / 360) = -1, so the principal is -45 - (-1 * 360) = 315 degrees. Listing additional coterminals is then just a matter of adding or subtracting 360 once, twice, three times, and so on.
Reference angles and quadrants
The reference angle is the acute angle (between 0 and 90 degrees, or 0 and pi/2 radians) formed between the terminal side and the nearest part of the x-axis. It is always positive and always at most 90 degrees. The rule is: in Quadrant I, the reference angle equals the principal angle itself. In Quadrant II, subtract from 180 degrees. In Quadrant III, subtract 180 degrees. In Quadrant IV, subtract from 360 degrees. Reference angles matter because the absolute value of any trig function equals the trig value of the reference angle, with only the sign changing by quadrant. This is the basis for the CAST rule (or ASTC rule): All trig functions are positive in Q I, Sine in Q II, Tangent in Q III, and Cosine in Q IV.
Degrees versus radians
Degrees divide a full rotation into 360 equal parts, a convention inherited from ancient Babylonian astronomy. Radians measure angles by the arc length they subtend on a unit circle, so a full rotation equals 2pi radians (approximately 6.2832). Both units describe the same angles; the choice is about context. Degrees are more intuitive for navigation, construction and everyday use. Radians are the natural unit for calculus and higher mathematics: derivatives of trig functions take their simplest forms only in radians, and Euler formula e^(i*theta) = cos(theta) + i*sin(theta) only holds cleanly in radians. To convert, use: radians = degrees * pi / 180, or degrees = radians * 180 / pi.
Common angles and their coterminal equivalents
| Angle (deg) | Principal (deg) | Reference (deg) | Quadrant |
|---|---|---|---|
| 0 | 0 | 0 | On axis |
| 30 | 30 | 30 | Q I |
| 45 | 45 | 45 | Q I |
| 60 | 60 | 60 | Q I |
| 90 | 90 | 0 | On axis |
| 120 | 120 | 60 | Q II |
| 135 | 135 | 45 | Q II |
| 150 | 150 | 30 | Q II |
| 180 | 180 | 0 | On axis |
| 210 | 210 | 30 | Q III |
| 225 | 225 | 45 | Q III |
| 240 | 240 | 60 | Q III |
| 270 | 270 | 0 | On axis |
| 300 | 300 | 60 | Q IV |
| 315 | 315 | 45 | Q IV |
| 330 | 330 | 30 | Q IV |
| 360 | 0 | 0 | On axis |
| 405 | 45 | 45 | Q I |
| 450 | 90 | 0 | On axis |
| 540 | 180 | 0 | On axis |
| 720 | 0 | 0 | On axis |
| -30 | 330 | 30 | Q IV |
| -45 | 315 | 45 | Q IV |
| -90 | 270 | 0 | On axis |
| -180 | 180 | 0 | On axis |
| -270 | 90 | 0 | On axis |
| -360 | 0 | 0 | On axis |
Standard angles in degrees, their principal coterminal form, reference angle, and quadrant. Use this as a quick lookup when reducing common trig values.
Frequently asked questions
What is a coterminal angle?
A coterminal angle is any angle that shares the same terminal side as a given angle when drawn in standard position (vertex at the origin, initial side along the positive x-axis). Two angles are coterminal when they differ by an integer multiple of 360 degrees or 2pi radians. For example, 30 degrees, 390 degrees and -330 degrees are all coterminal because they all end up pointing in the same direction.
How do you find positive and negative coterminal angles?
Add 360 degrees (or 2pi radians) repeatedly to produce positive coterminals, or subtract 360 degrees (or 2pi radians) repeatedly to produce negative coterminals. For an angle of 50 degrees, the positive coterminals include 410, 770 and 1130 degrees, while the negative coterminals include -310, -670 and -1030 degrees. There are infinitely many of each kind.
What is the principal coterminal angle?
The principal coterminal angle (sometimes called the reference coterminal or the reduced angle) is the unique coterminal angle that falls in the range [0, 360) degrees, or equivalently [0, 2pi) radians. It is found by repeatedly adding or subtracting 360 (or 2pi) until the result falls in that interval, which is the same as taking the non-negative remainder when dividing by 360.
Do coterminal angles have the same trig values?
Yes. The sine, cosine and tangent of coterminal angles are always equal, because those functions depend only on where the terminal side points, not on how many full rotations were made. sin(30 deg) = sin(390 deg) = sin(-330 deg), and the same holds for cosine and tangent. This is why coterminal reduction, finding the principal angle, is a standard first step when evaluating trig functions of large or negative angles.
What is the difference between a coterminal angle and a reference angle?
Coterminal angles share the same terminal side as the original angle, so they are equal in a rotational sense, just reached by different numbers of revolutions. A reference angle is the acute angle (0 to 90 degrees) between the terminal side and the nearest part of the x-axis. The reference angle is always positive and at most 90 degrees, and it is used to evaluate trig functions by reducing any angle to a first-quadrant equivalent.
How do coterminal angles work in radians?
In radians, the period is 2pi (approximately 6.2832) instead of 360 degrees. Add or subtract any integer multiple of 2pi to get a coterminal angle. The principal coterminal angle lies in [0, 2pi). For example, 7pi/4 is already in range, but 9pi/4 reduces to pi/4 (since 9pi/4 - 2pi = pi/4). Converting to radians first and then applying the same modulo logic gives the same result.