Coterminal Angle Calculator

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°