Double Angle and Half Angle Calculator

How it works

sin 2θ = 2 sin θ cos θ · cos 2θ = cos²θ − sin²θ = 1 − 2 sin²θ = 2 cos²θ − 1 · tan 2θ = 2 tan θ / (1 − tan²θ) · sin(θ/2) = ±√((1 − cos θ)/2) · cos(θ/2) = ±√((1 + cos θ)/2) · tan(θ/2) = sin θ / (1 + cos θ)

The double-angle identities come from the sum formulas with both angles equal, and the half-angle identities come from solving the cos 2θ identity for the smaller angle. The ± sign in a half-angle formula is set by the quadrant of θ/2, not θ. When you know only one ratio, the quadrant fixes the signs of the others (for example sin θ = 3/5 in quadrant II forces cos θ = −4/5). The calculator rebuilds θ from that information and evaluates every identity, showing fractions when the value is an exact fraction (OpenStax Precalculus §7.3).

Worked example

θ = 30°: sin 2θ = sin 60° = 2 · (1/2) · (√3/2) ≈ 0.866025; cos 2θ = 1 − 2 · (1/2)² = 1/2 = 0.5; tan 2θ ≈ 1.732051. Half angle: θ/2 = 15° is in quadrant I, so sin 15° = √((1 − cos 30°)/2) ≈ 0.258819, cos 15° ≈ 0.965926 and tan 15° ≈ 0.267949.

Sources: OpenStax Precalculus 2e — 7.3 Double-Angle, Half-Angle, and Reduction Formulas · Paul’s Online Math Notes — Trig Formulas

Frequently asked questions