Sum-to-Product Formulas
Convert sums of trig functions into products.
Simplify expressions and solve equations in trigonometry and signal processing.
The Formula
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)
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)
These identities convert a sum (or difference) of two trig functions into a product. They are particularly useful for solving equations and analyzing wave interference.
Variables
| Symbol | Meaning |
|---|---|
| A, B | Any two angles |
| (A+B)/2 | Average of the two angles |
| (A-B)/2 | Half the difference of the two angles |
Example 1
Simplify sin(75°) + sin(15°)
= 2 sin((75+15)/2) cos((75-15)/2)
= 2 sin(45°) cos(30°)
= 2 × (√2/2) × (√3/2)
= √6/2 ≈ 1.2247
Example 2
Simplify cos(5x) - cos(3x)
= -2 sin((5x+3x)/2) sin((5x-3x)/2)
= -2 sin(4x) sin(x)
When to Use It
Use sum-to-product formulas when:
- Solving trigonometric equations by factoring
- Analyzing beat frequencies in acoustics (wave interference)
- Simplifying trigonometric expressions in calculus
- Working with signal processing and Fourier analysis