Sum-to-Product & Product-to-Sum Formulas
Turning sums into products and products into sums — the bridge between addition and factoring in trigonometry
The Big Idea
You already know the angle-addition formulas. Nothing new is postulated here. Every formula in this lesson is derived by adding or subtracting two addition identities you already have. The payoff is enormous: you gain the ability to factor trigonometric sums (essential for solving equations) and to linearize trigonometric products (essential for integration and simplification).
1. Product-to-Sum Formulas
These formulas rewrite a product of two trigonometric functions as a sum or difference. There are four of them:
- $\sin A \cos B = \dfrac{1}{2}\bigl[\sin(A+B) + \sin(A-B)\bigr]$
- $\cos A \sin B = \dfrac{1}{2}\bigl[\sin(A+B) - \sin(A-B)\bigr]$
- $\cos A \cos B = \dfrac{1}{2}\bigl[\cos(A-B) + \cos(A+B)\bigr]$
- $\sin A \sin B = \dfrac{1}{2}\bigl[\cos(A-B) - \cos(A+B)\bigr]$
Derivation (sin A cos B)
Start from the two addition formulas you already know:
Add equations (i) and (ii):
$$\sin(A+B) + \sin(A-B) = 2\sin A\cos B$$Divide both sides by 2:
$$\boxed{\sin A\cos B = \tfrac{1}{2}[\sin(A+B) + \sin(A-B)]}$$Derivation (sin A sin B)
Start from the cosine addition formulas:
Subtract (iv) from (iii):
$$\cos(A-B) - \cos(A+B) = 2\sin A\sin B$$Divide by 2: