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
| Field | What to enter |
|---|---|
| Mode | Choose one: Find coterminal angles, Check two angles |
| Angle α | Units: °, rad, π rad, grad, turn |
| Angle β | Units: °, 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°) (°) |
|---|---|
| 500 | 140 |
| 750 | 30 |
| 1,000 | 280 |
| 1,250 | 170 |
| 1,500 | 60 |
| 2,000 | 200 |
Formula
Coterminal angles: α + k·360° (k any integer)
In [0°, 360°): α mod 360°
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.
Frequently asked questions
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.