Sum-to-product identities · derivation ·proof

Where sin A + sin B = 2 sin((A + B)/2) cos((A − B)/2) comes from

Average and half-difference. The pattern to remember: sum-to-product always talks about the average of the two angles and half their difference. Physically, that is the carrier frequency and the beat.

Building it step by step

1. Take u = (A + B)/2, v = (A − B)/2
   Change of variables.
2. Expand sin(u ± v)
   Sum and difference formulas.
3. Add
   Odd terms cancel, even terms double.
4. 2 sin u cos v
   Result.

How to recall it under pressure

Re-derive it from the first line rather than searching memory: start from sin(a) + sin(b) and do the one algebraic move the derivation above makes. If you need the statement itself, it is on the sum to product: sin a + sin b page.

sampled at 800 random anglesSource: OpenStax Precalculus, Ch. 7.2–7.3 · Paul's Online Math Notes · revised 2026-09-26 ·how we check ·accuracy policy ·report an error