Identity library · 4 formulas
Sum-to-product identities
The reverse direction. Solving sin A + sin B = 0 or factoring a trig expression is where these pay off, because an equation with one product equal to zero can be split into two simple equations.
Every sum-to-product identities on this site
| Formula | Name | Used for |
|---|---|---|
| sin A + sin B = 2 sin((A + B)/2) cos((A − B)/2) | Sum to product: sin A + sin B | solve, simplify, convert |
| sin A − sin B = 2 cos((A + B)/2) sin((A − B)/2) | Sum to product: sin A − sin B | solve, simplify, convert |
| cos A + cos B = 2 cos((A + B)/2) cos((A − B)/2) | Sum to product: cos A + cos B | simplify, convert, solve |
| cos A − cos B = −2 sin((A + B)/2) sin((A − B)/2) | Sum to product: cos A − cos B | simplify, convert, solve |
Sum-to-product identities (4)
Sums and differences rewritten as a single product.
| Identity | Name | proof |
|---|---|---|
| sin A + sin B = 2 sin((A + B)/2) cos((A − B)/2) | Sum to product: sin A + sin B | proof |
| sin A − sin B = 2 cos((A + B)/2) sin((A − B)/2) | Sum to product: sin A − sin B | proof |
| cos A + cos B = 2 cos((A + B)/2) cos((A − B)/2) | Sum to product: cos A + cos B | proof |
| cos A − cos B = −2 sin((A + B)/2) sin((A − B)/2) | Sum to product: cos A − cos 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.