Product-to-Sum Formulas
Convert products of trig functions into sums.
Essential for integration, signal processing, and wave analysis.
The Formula
sin A × cos B = ½[sin(A+B) + sin(A-B)]
cos A × cos B = ½[cos(A-B) + cos(A+B)]
sin A × sin B = ½[cos(A-B) - cos(A+B)]
cos A × cos B = ½[cos(A-B) + cos(A+B)]
sin A × sin B = ½[cos(A-B) - cos(A+B)]
These identities rewrite a product of two trig functions as a sum or difference. They are the reverse of the sum-to-product formulas and are essential for integration.
Variables
| Symbol | Meaning |
|---|---|
| A, B | Any two angles |
| A+B | Sum of the angles |
| A-B | Difference of the angles |
Example 1
Rewrite sin(3x)cos(x) as a sum
sin(3x)cos(x) = ½[sin(3x+x) + sin(3x-x)]
= ½[sin(4x) + sin(2x)]
Example 2
Evaluate cos(75°)cos(15°)
= ½[cos(75°-15°) + cos(75°+15°)]
= ½[cos(60°) + cos(90°)]
= ½[0.5 + 0]
= 0.25
When to Use It
Use product-to-sum formulas when:
- Integrating products of sine and cosine functions
- Analyzing modulated signals in communications engineering
- Simplifying complex trigonometric expressions
- Converting between frequency-domain and time-domain representations