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
Limitations
- These identities apply only to sin+sin, sin−sin, cos+cos, and cos−cos — there is no sum-to-product form for sin±cos directly
- The conversion turns a sum into a product, which is useful for factoring and solving equations but may be harder to evaluate numerically
- In acoustics, the beat frequency between two tones uses the cos−cos form: two close frequencies produce a slow amplitude envelope at their difference frequency
Key Notes
- Core formulas: sin A + sin B = 2 sin½(A+B) cos½(A−B); sin A − sin B = 2 cos½(A+B) sin½(A−B); cos A + cos B = 2 cos½(A+B) cos½(A−B); cos A − cos B = −2 sin½(A+B) sin½(A−B).
- Derived from angle addition formulas: These identities are found by adding or subtracting the sin(P+Q) and sin(P−Q) expansions, then substituting A = P+Q and B = P−Q. They are the inverse of the product-to-sum identities.
- Simplifies integration of sums: Integrating sin(3x) + sin(7x) directly requires two steps; converting to a product first can simplify the work, especially in Fourier analysis.
- Beat frequencies in physics: Two sound waves of slightly different frequencies f₁ and f₂ combine to produce a beating wave: cos(2πf₁t) + cos(2πf₂t) = 2 cos(π(f₁+f₂)t) cos(π(f₁−f₂)t). The slow oscillation at (f₁−f₂)/2 is the audible beat.
- Applications: Sum-to-product identities appear in signal processing (modulation and demodulation), acoustic interference analysis, and solving trigonometric equations where a sum of trig functions equals a constant.