Double Angle vs Half Angle Formulas

Double angles merge two equal angles into one: θ + θ = 2θ. Half angles split θ into two equal parts: θ/2 and θ/2. The algebra mirrors that language. Double-angle identities expand the argument; half-angle identities compress it.

Key difference: the argument inside the function doubles in one family and halves in the other. Pick the identity that matches what you see in the problem, not the one you memorized most recently.

Double-angle formulas often appear when a product sin θ cos θ or an equation with sin(2θ) needs simplification. Half-angle formulas often appear when you need sin(θ/2) or cos(θ/2) from a value at θ, especially in integration or root-finding on the unit circle.

Both families connect through the Pythagorean identity. Starting from cos(2θ) = 1 − 2 sin² θ and replacing θ with θ/2 gives a half-angle form for sin²(θ/2). That link explains why textbooks place the two topics back to back.

Double angle (sine): sin(2θ) = 2 sin θ cos θ

Double angle (cosine examples): cos(2θ) = cos² θ − sin² θ; cos(2θ) = 2 cos² θ − 1

Half angle (examples):

• sin²(θ/2) = (1 − cos θ) / 2
• cos²(θ/2) = (1 + cos θ) / 2
• tan(θ/2) = sin θ / (1 + cos θ)

The full double-angle list with usage notes lives in Double Angle Formula. Half-angle square forms often include a ± sign when you take square roots; double-angle identities do not require that extra branch choice when used forward.

1. Read the angle inside the function. Is it 2θ, θ, or θ/2?
2. If you see 2θ, use double-angle identities. Match sin, cos, or tan to the correct form from the double-angle list.
3. If you see θ/2, use half-angle identities. Watch for squared forms and sign choices when taking square roots.
4. Verify numerically when possible. Pick a concrete θ and compare both paths.
5. Cross-check algebra-heavy steps. When in doubt, confirm the double-angle side with Double Angle Identities examples before mixing in half-angle steps.

Double-angle setup: Find sin(70°) given sin(140°) and cos(70°).

sin(140°) = 2 sin(70°) cos(70°). This is a double-angle setup, not a half-angle one, because 140° = 2 × 70°.

Half-angle setup: Find sin(15°) given cos(30°).

sin²(15°) = (1 − cos(30°)) / 2 = (1 − √3/2) / 2. Take the positive root for the first-quadrant angle.

Contrast: cos(60°) = 2 cos²(30°) − 1 uses double angle. cos²(15°) = (1 + cos(30°)) / 2 uses half angle. Same underlying identity chain, different direction.