Coterminal Angle Calculator

What Are Coterminal Angles? Two angles are coterminal if they share the same initial side (the starting ray) and the same terminal side (the ending ray). They differ only by whole rotations of 360°, or 2π radians. A wheel at 30° and the same wheel one full turn later at 390° point in exactly the same direction, and every trig function agrees they are the same angle.

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°