Calculator Input
Example Data Table
| Input | Unit | Smallest Positive Coterminal Angle | Quick Reason |
|---|---|---|---|
| 725 | Degrees | 5° | 725 - 2×360 = 5 |
| -45 | Degrees | 315° | -45 + 360 = 315 |
| 720 | Degrees | 360° | Exact full turns report the positive turn. |
| 11*pi/3 | Radians | 5π/3 | 11π/3 - 2π = 5π/3 |
| -13*pi/6 | Radians | 11π/6 | -13π/6 + 4π = 11π/6 |
Formula Used
Degrees
General coterminal family: θ + 360k, where k is any integer.
Smallest positive coterminal angle: θ₊ = ((θ mod 360) + 360) mod 360. If the result is 0, report 360°.
Radians
General coterminal family: θ + 2πk, where k is any integer.
Smallest positive coterminal angle: θ₊ = ((θ mod 2π) + 2π) mod 2π. If the result is 0, report 2π.
The calculator first evaluates the entered expression, applies the correct cycle length, and then converts exact multiples of a full turn into a strictly positive answer.
How to Use This Calculator
- Enter an angle as a number or expression, such as
725,-45, or11*pi/3. - Select the input unit: degrees or radians.
- Choose how many decimal places you want in the final output.
- Set how many coterminal angles should appear in the generated table.
- Choose how many points should appear on the Plotly graph.
- Press the calculate button to show the result above the form.
- Use the export buttons to download the summary as CSV or PDF.
FAQs
1) What is a positive coterminal angle?
It is an angle greater than zero that ends at the same terminal side as the original angle. Coterminal angles differ by whole revolutions only.
2) Why does 720° return 360° here?
This page reports the smallest strictly positive coterminal angle. Since 720° is two exact turns, the positive representative becomes one full positive turn, or 360°.
3) Can I enter negative angles?
Yes. Negative inputs are supported. The calculator adds full turns until the result becomes positive while preserving the same terminal side.
4) Can I work in radians?
Yes. Use the radians option and enter decimal values or simple expressions containing pi, such as -13*pi/6 or 7*pi.
5) What does the value k mean?
The integer k counts how many full turns were added or removed. In degrees, each turn is 360. In radians, each turn is 2π.
6) What is the difference between remainder and positive coterminal angle?
A non-negative remainder may be zero. A positive coterminal angle must be greater than zero, so exact multiples of a full turn are reported as one full positive cycle.
7) Why does the graph rise in equal steps?
Every coterminal angle in the plotted sequence is separated by one complete revolution. That constant interval creates a straight, evenly spaced upward pattern.
8) Where are coterminal angles useful?
They are useful in trigonometry, unit-circle work, geometry, navigation, rotations, engineering motion, and simplifying repeated angular positions in periodic systems.