Coterminal Angle Calculator
Find coterminal angles for any degree or radian measure.
Calculate positive and negative coterminals, reference angles, and quadrant location with full explanation.
What Are Coterminal Angles? Two angles are coterminal if they share the same terminal side when drawn in standard position. Standard position: vertex at the origin, initial side along the positive x-axis, rotating counterclockwise. Coterminal angles differ by any integer multiple of 360° (or 2π radians). Every angle has infinitely many coterminal angles — both positive and negative.
Finding Coterminal Angles For an angle θ in degrees, any coterminal angle = θ + 360°·n where n is any integer. For radians: θ + 2π·n. To find the principal coterminal angle (in [0°, 360°)): If θ > 360°: subtract 360° until in range. If θ < 0°: add 360° until in range. In one step: θ_principal = θ mod 360° (using mathematical modulo).
Reference Angle The reference angle is the acute angle between the terminal side and the x-axis. It is always between 0° and 90° (0 and π/2 radians), and always positive. Quadrant I (0°–90°): reference = θ Quadrant II (90°–180°): reference = 180° − θ Quadrant III (180°–270°): reference = θ − 180° Quadrant IV (270°–360°): reference = 360° − θ Reference angles are used to evaluate trig functions: sin(150°) = sin(30°) = 1/2.
The Four Quadrants Quadrant I (0°–90°): sin+, cos+, tan+. All trig functions positive. Quadrant II (90°–180°): sin+, cos−, tan−. “All Students Take Calculus” mnemonic. Quadrant III (180°–270°): sin−, cos−, tan+. Quadrant IV (270°–360°): sin−, cos+, tan−. Quadrantal angles (0°, 90°, 180°, 270°) lie on axes, not in any quadrant.
Radians and Degrees π radians = 180°. To convert: degrees × π/180 = radians. Radians × 180/π = degrees. Common conversions: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 180° = π, 270° = 3π/2. The radian is the SI unit for angle — defined as the angle subtended by an arc equal in length to the radius.
Applications Coterminal angles appear throughout trigonometry, calculus, and physics. Periodic functions (sin, cos) repeat every 2π — coterminal inputs give identical outputs. In signal processing, phase angles wrap around (a phase of 370° = 10°). Angular position in robotics and navigation always needs modular reduction. In aviation, compass bearings are in [0°, 360°) — adding/subtracting turns uses coterminal logic.
Negative Angles Negative angles rotate clockwise. −90° is the same terminal position as 270°. Calculator tip: JavaScript % operator gives negative results for negative inputs. True mathematical modulo: ((θ % 360) + 360) % 360 gives the correct principal angle.