# Coterminal Angle Calculator

URL: https://speedycalculator.com/calculators/math/coterminal-angle/
Last updated: 2026-10-30

> The coterminal angle calculator finds the coterminal angle between 0° and 360° (or 0 and 2π), positive and negative coterminal angles, or checks whether two angles are coterminal.

## Inputs

- Mode
- Angle α (°, rad, π rad, grad, turn)
- Angle β (°, rad, π rad, grad, turn)

## Results

- Coterminal angle in [0°, 360°)
- Positive coterminal angle
- Negative coterminal angle
- Full turns removed
- Quadrant
- Are they coterminal?
- Difference in full turns

## Formula

```
Coterminal angles: α + k·360° (k any integer)
In [0°, 360°): α mod 360°
```

## Explanation

Coterminal angles share the same terminal side: 30°, 390° and −330° all point the same way. They have identical trigonometric values, so reducing an angle to [0°, 360°) is the first step in evaluating sin 1000° or cos(−750°).

### How to use the Coterminal Angle Calculator

- **Mode**
- **Angle α**
- **Angle β**

Results update as you type, and everything is calculated in your browser.

### How it works

Choose the mode. The calculator reduces the angle with the modulo operation and lists the nearest positive and negative coterminal angles.

### Tips

- Angles are measured counter-clockwise from the positive x-axis by default.
- Add or subtract 360° (2π) to move between coterminal angles.
- Complementary angles add to 90°, supplementary angles to 180°.
- Use the unit menu to switch between degrees, radians and gradians.

### Common mistakes to avoid

- Mixing up complementary and supplementary angles.
- Forgetting that a reference angle is always between 0° and 90°.
- Leaving an angle in radians when the question asks for degrees.

### Inputs explained

FieldWhat to enterModeChoose one: Find coterminal angles, Check two anglesAngle αUnits: °, rad, π rad, grad, turnAngle βUnits: °, rad, π rad, grad, turn

### What each result means

- **Coterminal angle in [0°, 360°)** = mod([Angle α], 360)
- **Positive coterminal angle** = mod([Angle α], 360) + 360
- **Negative coterminal angle** = mod([Angle α], 360) - 360
- **Full turns removed** = floor([Angle α]÷360)
- **Difference in full turns** = ([Angle α] - [Angle β])÷360

### Example values

How the main result changes when **Angle α** changes and everything else stays at the example values above:

Angle α (°)Coterminal angle in [0°, 360°) (°)500140750301,0002801,2501701,500602,000200

## Worked example

Inputs — Mode: Find coterminal angles; Angle α: 1,000 °.

Results — **Coterminal angle in [0°, 360°) = 280 °**; Positive coterminal angle = 640 °; Negative coterminal angle = -80 °; Full turns removed = 2; Quadrant = Quadrant IV.

## Key terms

- **Terminal side**: The final ray of an angle in standard position.

## FAQ

**What are coterminal angles?**
Angles with the same initial and terminal side, differing by multiples of 360°.

**How do I find a coterminal angle?**
Add or subtract 360° (or 2π) as many times as needed.

**Do coterminal angles have the same sine?**
Yes – all trig functions are equal.

**Is 0° coterminal with 360°?**
Yes.


## Sources

- [OpenStax – Prealgebra 2e](https://openstax.org/details/books/prealgebra-2e)
- [OpenStax – Algebra and Trigonometry 2e](https://openstax.org/details/books/algebra-and-trigonometry-2e)
- [Wolfram MathWorld – Coterminal angle](https://mathworld.wolfram.com/CoterminalAngle.html)