Quick answer
Coterminal angles differ by multiples of 360°: θ ± 360°·k. An angle of 400° is coterminal with 40° (the smallest non-negative one) and with 760°.
Enter an angle in degrees. The calculator returns the smallest non-negative coterminal angle and the angles one turn either side.
Related tools: Reference Angle Calculator: Acute angle between the terminal side and the x-axis. Angle of Elevation Calculator: Angle of elevation from a height difference and horizontal distance. Angle of Depression Calculator: Angle of depression from a height difference and horizontal distance.
How to use the Coterminal Angle Calculator
- Enter angle (°).
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Angle | A number, in ° | — |
Formula and method
Formula
θ ± 360°·k
IEEE-754 double precision; shown to at most 6 decimal places.
Adds and subtracts full turns; also gives the equivalent angle in [0°, 360°).
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Angle = 400.
| Result | Value |
|---|---|
| Smallest non-negative coterminal angle | 40° |
| −360° turn | 40° |
| +360° turn | 760° |
Example 2
Inputs: Angle = -45.
| Result | Value |
|---|---|
| Smallest non-negative coterminal angle | 315° |
| −360° turn | -405° |
| +360° turn | 315° |
The first line is the headline result (smallest non-negative coterminal angle). Below it the coterminal Angle Calculator lists −360° turn and +360° turn, which describe the same case from other angles. IEEE-754 double precision; shown to at most 6 decimal places.
Same terminal side
Coterminal angles end at the same position on the unit circle, so they have identical sine, cosine and tangent. 400° − 360° = 40°; −320° + 360° = 40° too.
Negative and large angles
For a negative angle add 360° until it is positive: −45° is coterminal with 315°. In radians, add or subtract 2π. Any real angle is accepted.
How the result changes with angle
Holding every other input at the first worked example, this table changes only angle. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| Angle | Smallest non-negative coterminal angle | −360° turn | +360° turn |
|---|---|---|---|
| 200 | 200° | -160° | 560° |
| 400 | 40° | 40° | 760° |
| 800 | 80° | 440° | 1,160° |
| 1600 | 160° | 1,240° | 1,960° |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Angle: left empty shows "Enter angle." beside that field and no result.
Choosing between the Coterminal Angle Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Reference Angle Calculator](/math/reference-angle/) | angle | Reference angle |
| [Angle of Elevation Calculator](/math/angle-of-elevation-calculator/) | height difference and horizontal distance | Angle of elevation |
| [Angle of Depression Calculator](/math/angle-of-depression-calculator/) | height difference and horizontal distance | Angle of depression |
| [Double Angle Calculator](/math/double-angle-calculator/) | angle and unit | sin 2θ |
| [Half Angle Calculator](/math/half-angle-calculator/) | angle and unit | sin(θ/2) |
Background: trigonometry
In a right triangle, sin θ is the opposite side over the hypotenuse, cos θ the adjacent side over the hypotenuse and tan θ the opposite over the adjacent. Their reciprocals are cosecant, secant and cotangent. At 30° the values are sin = 0.5, cos = 0.866025 and tan = 0.577350.
Check the angle unit before you calculate: degrees convert to radians by multiplying by π/180, so 30° is about 0.5236 rad. Some values are undefined, for example tan 90°, because the cosine is zero there and the ratio would divide by zero.
Assumptions and limitations
- Any real angle
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for angle to confirm this.
Frequently asked questions
What is coterminal with 30°?
390° (30° + 360°) and −330° (30° − 360°) are coterminal with 30°.
How do I reduce 1000°?
Subtract 720° to get 280°.
Do coterminal angles have the same trig values?
Yes, they end at the same point on the unit circle, so sine, cosine and tangent are equal.
What does the Coterminal Angle Calculator calculate?
Angles sharing the same terminal side. It returns Smallest non-negative coterminal angle, −360° turn and +360° turn.
What formula does the Coterminal Angle Calculator use?
θ ± 360°·k. Adds and subtracts full turns; also gives the equivalent angle in [0°, 360°).
What do I need to enter in the coterminal Angle Calculator?
You enter Angle (°).
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. Any real angle.
What is an example of a coterminal Angle result?
With angle 400, the smallest non-negative coterminal angle is 40°.
Related calculators
- Reference Angle Calculator — Acute angle between the terminal side and the x-axis.
- Angle of Elevation Calculator — Angle of elevation from a height difference and horizontal distance.
- Angle of Depression Calculator — Angle of depression from a height difference and horizontal distance.
- Double Angle Calculator — Double-angle identities evaluated at an angle.
- Half Angle Calculator — Half-angle identities evaluated at an angle.
- Trigonometry Calculator — All six trigonometric functions of an angle.
- Cotangent Calculator — cot of an angle in degrees or radians.
Sources
- Standard trigonometric definitions on the unit circle (NIST DLMF §4.14)
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.