Identity library · 4 formulas
Product-to-sum identities
Four formulas that turn sin A cos B and friends into sums. Integral calculus is the main customer: ∫sin 3x cos x dx is not approachable until the product is a sum, and the same trick handles ∫sin²x dx through cos²x = (1+cos 2x)/2.
Every product-to-sum identities on this site
| Formula | Name | Used for |
|---|---|---|
| sin A cos B = ½[sin(A + B) + sin(A − B)] | Product to sum: sin A cos B | integrate, simplify, convert |
| cos A sin B = ½[sin(A + B) − sin(A − B)] | Product to sum: cos A sin B | integrate, simplify, convert |
| cos A cos B = ½[cos(A − B) + cos(A + B)] | Product to sum: cos A cos B | integrate, simplify, convert |
| sin A sin B = ½[cos(A − B) − cos(A + B)] | Product to sum: sin A sin B | integrate, simplify, convert |
Product-to-sum identities (4)
Products of sines and cosines rewritten as sums.
| Identity | Name | proof |
|---|---|---|
| sin A cos B = ½[sin(A + B) + sin(A − B)] | Product to sum: sin A cos B | proof |
| cos A sin B = ½[sin(A + B) − sin(A − B)] | Product to sum: cos A sin B | proof |
| cos A cos B = ½[cos(A − B) + cos(A + B)] | Product to sum: cos A cos B | proof |
| sin A sin B = ½[cos(A − B) − cos(A + B)] | Product to sum: sin A sin B | proof |
Doing something with them
- Which identity should I use? — describe an expression, get the rule.
- Verify — check a claim, see a counterexample if it is false.
- Printable cheat sheet — this family pre-selected.