Coterminal Angle Calculator
Find coterminal angles, reference angles, quadrant location, and trig values for any angle in degrees or radians.
Shows positive and negative coterminals.
What Are Coterminal Angles? Two angles are coterminal if they share the same initial side (starting ray) and the same terminal side (ending ray) — they simply differ by full rotations of 360° (or 2π radians).
Finding Coterminal Angles Coterminal Angle = θ + 360° × n (for any integer n)
In radians: Coterminal Angle = θ + 2π × n
Positive coterminals: add 360°, 720°, 1080°… Negative coterminals: subtract 360°, 720°, 1080°…
Standard Position Angle To find the standard position angle (always between 0° and 360°): θ_standard = θ mod 360° (handling negatives: add 360° until positive)
In radians: θ_standard = θ mod 2π
Reference Angle The reference angle is the acute angle (0° to 90°) between the terminal side and the x-axis.
- Quadrant I: reference = θ
- Quadrant II: reference = 180° − θ
- Quadrant III: reference = θ − 180°
- Quadrant IV: reference = 360° − θ
Quadrant Identification
- Quadrant I: 0° < θ < 90°
- Quadrant II: 90° < θ < 180°
- Quadrant III: 180° < θ < 270°
- Quadrant IV: 270° < θ < 360°
- On an axis: 0°, 90°, 180°, 270°, 360° are quadrantal angles
Degree to Radian Conversion Radians = Degrees × (π / 180) Degrees = Radians × (180 / π)
Applications Coterminal angles are used extensively in:
- Trigonometric function evaluation (sin, cos, tan repeat every 360°)
- Polar coordinates and complex number arguments
- Physics: angular position of rotating objects (wheels, rotors, Earth)
- Signal processing: phase angles that wrap around 360°