Lesson 4: Product-to-Sum and Sum-to-Product Formulas
Estimated time: 30-35 minutes
Learning Objectives
- State all four product-to-sum formulas
- State all four sum-to-product formulas
- Convert products of trig functions into sums
- Convert sums of trig functions into products
Product-to-Sum Formulas
cos A cos B = (1/2)[cos(A − B) + cos(A + B)]
sin A sin B = (1/2)[cos(A − B) − cos(A + B)]
sin A cos B = (1/2)[sin(A + B) + sin(A − B)]
cos A sin B = (1/2)[sin(A + B) − sin(A − B)]
Example 1
Write cos 3x cos 5x as a sum.
= (1/2)[cos(3x−5x) + cos(3x+5x)] = (1/2)[cos(−2x) + cos 8x] = (1/2)(cos 2x + cos 8x)
Example 2
Write sin 7x sin 3x as a sum.
= (1/2)[cos(7x−3x) − cos(7x+3x)] = (1/2)(cos 4x − cos 10x)
Sum-to-Product Formulas
sin A + sin B = 2 sin((A+B)/2) cos((A−B)/2)
sin A − sin B = 2 cos((A+B)/2) sin((A−B)/2)
cos A + cos B = 2 cos((A+B)/2) cos((A−B)/2)
cos A − cos B = −2 sin((A+B)/2) sin((A−B)/2)
Example 3
Write sin 5x + sin 3x as a product.
= 2 sin((5x+3x)/2) cos((5x−3x)/2) = 2 sin 4x cos x
Example 4
Write cos 6x − cos 2x as a product.
= −2 sin((6x+2x)/2) sin((6x−2x)/2) = −2 sin 4x sin 2x
Check Your Understanding
1. Write sin 4x cos 2x as a sum.
2. Write cos 8x + cos 2x as a product.
3. Write sin 3x − sin x as a product.
Key Takeaways
- Product-to-sum formulas convert products into sums (useful in calculus integration).
- Sum-to-product formulas convert sums into products (useful for solving equations).
- These formulas are derived from the sum/difference formulas by adding/subtracting them.