What is a Coterminal Angle?
Coterminal angles are angles that share the same terminal side when drawn in standard position. Standard position means the initial side lies along the positive x-axis and the vertex is at the origin. Two different angles can look identical on a coordinate plane — that makes them coterminal.
In other words, two angles are coterminal when they differ by one or more full rotations (360° or 2π radians). For example, 45°, 405°, and −315° are all coterminal because they all end at the same ray.
Coterminal Angles Formula
To find a coterminal angle of a given angle θ, add or subtract any whole-number multiple of a full rotation:
- Degrees: θ ± 360° × n, where n is any integer (…−2, −1, 0, 1, 2, …)
- Radians: θ ± 2π × n, where n is any integer
Two angles α and β are coterminal if and only if α − β = 360° × n (or 2πn in radians) for some integer n.
How to Find a Coterminal Angle Between 0° and 360°
To find the unique coterminal angle in the range [0°, 360°), use the modulo operation:
- Compute normalized = θ mod 360° (keep the remainder when dividing by 360)
- If the result is negative, add 360°
- The result is the principal (reference) coterminal angle between 0° and 360°
Example: Find the coterminal angle of −120° in [0°, 360°):
- −120 mod 360 = 240 → the answer is 240°
For radians, the same logic applies using 2π (≈ 6.2832 rad) instead of 360°.
Positive and Negative Coterminal Angles
Positive coterminal angles are found by adding multiples of 360°:
- θ + 360° (n = 1)
- θ + 720° (n = 2)
- θ + 1080° (n = 3)
- … and so on
Negative coterminal angles are found by subtracting multiples of 360°:
- θ − 360° (n = −1)
- θ − 720° (n = −2)
- θ − 1080° (n = −3)
- … and so on
Common Coterminal Angle Examples
| Angle | Normalized [0°, 360°) | +1 Coterminal | −1 Coterminal |
|---|---|---|---|
| 0° | 0° | 360° | −360° |
| 45° | 45° | 405° | −315° |
| 90° | 90° | 450° | −270° |
| 180° | 180° | 540° | −180° |
| 270° | 270° | 630° | −90° |
| −30° | 330° | 690° | −390° |
| −90° | 270° | 630° | −450° |
| π/4 rad | π/4 rad | 9π/4 rad | −7π/4 rad |
| π rad | π rad | 3π rad | −π rad |
Coterminal Angles vs Reference Angles
These two concepts are often confused:
- Coterminal angles share the same terminal side — they can be any angle ±360°×n away from the original.
- Reference angle is the acute angle between the terminal side and the x-axis — it is always between 0° and 90°.
Coterminal Angles in Degrees and Radians
The same rules apply in both unit systems — just replace 360° with 2π rad:
- Degrees: add/subtract 360° per full rotation
- Radians: add/subtract 2π (≈ 6.2832) per full rotation
To convert between systems: 1° = π/180 rad and 1 rad = 180°/π ≈ 57.2958°.
Frequently Asked Questions
Can there be infinitely many coterminal angles?
Yes — there are infinitely many coterminal angles for any given angle, because you can always add or subtract another 360° (or 2π).
Is 0° coterminal with 360°?
Yes. 360° − 0° = 360° = 360° × 1, so they are coterminal. In fact, any multiple of 360° is coterminal with 0°.
Are −45° and 315° coterminal?
Yes. 315° − (−45°) = 360° = 360° × 1, so they are coterminal angles.
How do I use this calculator?
Select a unit (degrees or radians) and a mode:
- Find coterminal angles: Enter any angle and get the normalized [0°, 360°) form plus a list of positive and negative coterminal angles.
- Check if two angles are coterminal: Enter two angles and the tool verifies whether they are coterminal by checking if their difference is a multiple of 360° (or 2π).