Product-to-sum identities · trigonometric identity
cos A sin B = ½[sin(A + B) − sin(A − B)] — Product to sum: cos A sin B
When to use it
Same family, reversed order — and this is where the sign flips. If your product is cos A sin B, the difference of sines is what you need.
Why it is true
Subtract the two sine expansions: sin(A + B) − sin(A − B) = 2 cos A sin B.
The full line-by-line version is on the proof page for product to sum: cos a sin b; the “how would I find this myself” version is in the derivation.
Where it comes from
Track which term cancels. Addition keeps the terms that appear with the same sign in both expansions; subtraction keeps the ones that appear with opposite signs.
- Expand bothStandard move.
- Want cos A sin B? It has opposite signsSo subtract.
- ½[sin(A + B) − sin(A − B)]Result.
Worked examples
No worked example is written for this identity yet —the examples index lists what is covered, and the pattern below shows the identity working at real angles.
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° | 0.218809 | 0.218809 | yes |
| 15° | 0.274338 | 0.274338 | yes |
| 18° | 0.293893 | 0.293893 | yes |
| 22.5° | 0.31977 | 0.31977 | yes |
| 30° | 0.352244 | 0.352244 | yes |
| 37° | 0.368769 | 0.368769 | yes |
| 45° | 0.369462 | 0.369462 | yes |
| 53° | 0.349476 | 0.349476 | yes |
| 60° | 0.31466 | 0.31466 | yes |
| 67.5° | 0.259766 | 0.259766 | yes |
| 75° | 0.187741 | 0.187741 | yes |
| 90° | 0 | 0 | yes |
| 120° | -0.46679 | -0.46679 | yes |
| 135° | -0.687569 | -0.687569 | yes |
| 150° | -0.861281 | -0.861281 | yes |
| 180° | -0.987688 | -0.987688 | yes |
| 210° | -0.791154 | -0.791154 | yes |
| 240° | -0.388573 | -0.388573 | yes |
| 270° | -0 | -0 | yes |
| 300° | 0.179184 | 0.179184 | yes |
| 330° | 0.090524 | 0.090524 | yes |
| 360° | -0.156434 | -0.156434 | yes |
The mistake students make
Using the sin A cos B formula (with a plus) for cos A sin B. Swapping which factor carries the plus is the whole content of this record.
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 product-to-sum identities.
Related identities
- sin A cos B = ½[sin(A + B) + sin(A − B)]Product to sum: sin A cos Bdetailsproof
- cos A cos B = ½[cos(A − B) + cos(A + B)]Product to sum: cos A cos Bdetailsproof
- sin A + sin B = 2 sin((A + B)/2) cos((A − B)/2)Sum to product: sin A + sin Bdetailsproof
Category hub: Product-to-sum identities · all identities: /identities