Reference Angle Calculator

How it works

Reduce θ to [0°, 360°) · Quadrant I: θ′ = θ · II: θ′ = 180° − θ · III: θ′ = θ − 180° · IV: θ′ = 360° − θ (in radians use π and 2π)

The reference angle is the acute angle (between 0° and 90°) that the terminal side of θ makes with the x-axis. First reduce θ to its coterminal angle between 0° and 360°, then apply the rule for its quadrant. Trig values of θ equal those of the reference angle up to a sign, and the sign comes from the quadrant (All Students Take Calculus: all positive in I, sine in II, tangent in III, cosine in IV).

Worked example

For θ = 225°: it is already between 0° and 360° and lies in Quadrant III, so θ′ = 225° − 180° = 45° (π/4 ≈ 0.785398 rad). In Quadrant III sine and cosine are negative and tangent is positive, so sin 225° = −sin 45° and tan 225° = tan 45° = 1.

Sources: Reference angles — OpenStax Precalculus 7.3 (Unit Circle) · Reference angle — Wikipedia

Frequently asked questions