Coterminal Angle Calculator
This calculator returns the reference angle for any coterminal angle: enter one angle and get its smallest positive coterminal angle, a negative coterminal angle, and the shared reference angle, all updated as you type.
Accepts any value. Negative angles and angles beyond 360° work directly.
What are coterminal angles?
Coterminal angles are angles in standard position that share the same terminal side. Two angles can look completely different in number, one small, one huge, one negative, and still point in the exact same direction.
Every angle in standard position starts with its initial side on the positive x-axis. Rotating counterclockwise by a positive amount, or clockwise by a negative amount, sweeps the terminal side to some final direction. Once the terminal side completes a full 360° turn, it returns to where it started, so adding or subtracting any whole number of full turns never changes which direction the terminal side points. Angles related this way are coterminal.
Because coterminal angles share a terminal side, they also share the same reference angle and the same sine, cosine, and tangent values. This is the same underlying idea used throughout the reference angle calculator: direction matters more than the raw number of degrees.
Coterminal angle formula
Any whole number of turns
angle ± 360°n
Any whole number of turns
angle ± 2πn
Here n is any positive integer (1, 2, 3…). Adding 360°n rotates forward by whole turns; subtracting 360°n rotates backward. The smallest positive coterminal angle uses the smallest n that lands the result between 0° and 360°.
Positive coterminal: 100°
- Add one full turn: 100° + 360° = 460°
- 460° is a valid positive coterminal angle of 100°
- Reference angle: 100° is in Quadrant II, so 180° − 100° = 80°
Positive coterminal = 460°, reference angle = 80°
Negative coterminal: 100°
- Subtract one full turn: 100° − 360° = −260°
- −260° points in the same direction as 100°
- Reference angle stays 80°, since direction hasn't changed
Negative coterminal = −260°
Three worked examples
Negative starting angle: −200°
- Add 360° to get a positive angle: −200° + 360° = 160°
- 160° is the smallest positive coterminal angle
- Quadrant II, so reference angle = 180° − 160° = 20°
Smallest positive coterminal = 160°, reference angle = 20°
Radian angle: 5π/6
- Positive coterminal: 5π/6 + 2π = 17π/6
- Negative coterminal: 5π/6 − 2π = −7π/6
- 5π/6 is in Quadrant II, so reference angle = π − 5π/6 = π/6
Coterminal pair: 17π/6 and −7π/6, reference angle = π/6
Notice the pattern: whatever the starting angle looks like, negative, fractional, or expressed in radians, finding a coterminal angle only ever involves adding or subtracting full turns. The direction, quadrant, and reference angle never change.
Common coterminal pairs
Each row shows one angle with a positive and a negative coterminal partner. All three share the same terminal side.
| Angle | Positive coterminal | Negative coterminal |
|---|---|---|
| 0° | 360° | -360° |
| 30° | 390° | -330° |
| 45° | 405° | -315° |
| 60° | 420° | -300° |
| 90° | 450° | -270° |
| 120° | 480° | -240° |
| 135° | 495° | -225° |
| 150° | 510° | -210° |
| 210° | 570° | -150° |
| 270° | 630° | -90° |
Coterminal angles and reference angles
A reference angle measures the acute distance between a terminal side and the x-axis. Since coterminal angles share the same terminal side by definition, they automatically share the same reference angle too. This is why the reference angle calculator always reduces an angle to its coterminal value between 0° and 360° as the very first step. Everything else, including the quadrant and the reference angle, follows from that single reduced value.
In practice this means you rarely need to memorize trigonometric values for huge angles like 750° or −480°. Find the coterminal angle between 0° and 360° first, then apply the ordinary quadrant formula. The result is identical to working with the original angle directly, just with smaller, easier numbers.
Frequently Asked Questions
What is a coterminal angle?
A coterminal angle is an angle that shares the same terminal side as another angle, even though its measure is different. Two angles are coterminal when they differ by a whole number of full turns, 360°n in degrees or 2πn in radians.
How do you find a positive coterminal angle?
Add 360° (or 2π radians) to the angle. If the angle is already negative or larger than 360°, keep adding or subtracting 360° until the result is the smallest positive coterminal angle, between 0° and 360°. This works because each 360° rotation returns the terminal side to the same direction, so repeating the operation never changes which angle it represents.
How do you find a negative coterminal angle?
Subtract 360° (or 2π radians) from the angle. This rotates one full turn backward and lands on a negative angle that still points in the same direction. This method is useful whenever a problem specifically asks for a negative coterminal angle rather than the smallest positive one.
Can two angles be coterminal and have different reference angles?
No. Coterminal angles always share the exact same terminal side, so they always share the exact same reference angle. If two angles have different reference angles, they cannot be coterminal. Checking whether two angles share a reference angle is actually one of the fastest ways to test whether they are coterminal in the first place.
What is the coterminal angle of 390°?
390° is already coterminal with 30°, since 390° − 360° = 30°. The smallest positive coterminal angle of 390° is 30°, and a negative coterminal angle is 30° − 360° = −330°. All three angles, 390°, 30°, and −330°, point in exactly the same direction and share the same reference angle of 30°.
Are coterminal angles equal?
No. Coterminal angles are not numerically equal, since they have different degree or radian measures. They are equal only in direction: their terminal sides point the same way, so their trigonometric function values are identical. This is why coterminal angles are often described as equivalent rather than equal, since the numbers differ even though the geometry does not.