Sum-to-product identities · trigonometric identity
sin A + sin B = 2 sin((A + B)/2) cos((A − B)/2) — Sum to product: sin A + sin B
When to use it
You are solving sin A + sin B = 0 or simplifying a sum of two sines. One product equal to zero splits into two simple equations — that is the payoff.
Why it is true
Write A = (A+B)/2 + (A−B)/2 and B = (A+B)/2 − (A−B)/2, expand both with the sine sum/difference formulas and add: the mixed terms double.
The full line-by-line version is on the proof page for sum to product: sin a + sin b; the “how would I find this myself” version is in the derivation.
Where it 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.
- Take u = (A + B)/2, v = (A − B)/2Change of variables.
- Expand sin(u ± v)Sum and difference formulas.
- AddOdd terms cancel, even terms double.
- 2 sin u cos vResult.
Worked examples
- Solve sin x + sin 3x = 0 on [0, 2π)x = 0, π/2, π, 3π/2 (and 2π excluded from the interval)
Checked at these angles
Two sides of the identity evaluated with the same engine that runs the verifier. A single line disagreeing would break the build.
| Angle | Left side | Right side | Agree |
|---|---|---|---|
| 0° | 0.156434 | 0.156434 | yes |
| 7.5° | 0.351224 | 0.351224 | yes |
| 15° | 0.542834 | 0.542834 | yes |
| 18° | 0.618034 | 0.618034 | yes |
| 22.5° | 0.7288 | 0.7288 | yes |
| 30° | 0.906737 | 0.906737 | yes |
| 37° | 1.063564 | 1.063564 | yes |
| 45° | 1.229605 | 1.229605 | yes |
| 53° | 1.379338 | 1.379338 | yes |
| 60° | 1.495346 | 1.495346 | yes |
| 67.5° | 1.60268 | 1.60268 | yes |
| 75° | 1.6913 | 1.6913 | yes |
| 90° | 1.809017 | 1.809017 | yes |
| 120° | 1.799606 | 1.799606 | yes |
| 135° | 1.679477 | 1.679477 | yes |
| 150° | 1.494522 | 1.494522 | yes |
| 180° | 0.987688 | 0.987688 | yes |
| 210° | 0.413545 | 0.413545 | yes |
| 240° | -0.088879 | -0.088879 | yes |
| 270° | -0.412215 | -0.412215 | yes |
| 300° | -0.507657 | -0.507657 | yes |
| 330° | -0.395472 | -0.395472 | yes |
| 360° | -0.156434 | -0.156434 | yes |
The mistake students make
Keeping two terms. The point of sum-to-product is that the right side is a single product — if your answer still has a '+', the reduction is unfinished.
Try it
- Verify this identity — both sides are pre-filled; change one character and watch the counterexample appear.
- Ask “which identity should I use?” about the left-hand side — this identity is what the chooser should rank first.
- Cheat sheet entry for sum-to-product identities.
Related identities
- sin A − sin B = 2 cos((A + B)/2) sin((A − B)/2)Sum to product: sin A − sin Bdetailsproof
- cos A + cos B = 2 cos((A + B)/2) cos((A − B)/2)Sum to product: cos A + cos Bdetailsproof
- sin A cos B = ½[sin(A + B) + sin(A − B)]Product to sum: sin A cos Bdetailsproof
Category hub: Sum-to-product identities · all identities: /identities