Sum to Product Calculator
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Understanding Sum to Product Formulas
What are Sum to Product Formulas?
Sum to product formulas convert the sum or difference of trigonometric functions into products, making many calculations simpler:
sin(A) + sin(B) = 2sin((A+B)/2)cos((A-B)/2)
sin(A) - sin(B) = 2cos((A+B)/2)sin((A-B)/2)
cos(A) + cos(B) = 2cos((A+B)/2)cos((A-B)/2)
cos(A) - cos(B) = -2sin((A+B)/2)sin((A-B)/2)
Derivation and Proof
These formulas can be derived using:
- Euler's formula: e^(ix) = cos(x) + isin(x)
- Half-angle formulas
- Addition and subtraction formulas
- Geometric proofs using unit circle
Important Properties
Symmetry
Results are independent of order for sums
Domain
Valid for all real numbers
Range
Depends on specific formula
Periodicity
Preserves periodic nature
Special Cases
| Angles (A,B) | sin(A) + sin(B) | cos(A) + cos(B) | Simplified Result |
|---|---|---|---|
| 0°, 0° | 0 | 2 | 2cos(0°) |
| 30°, 60° | 1.5 | 1 | 2sin(45°)cos(15°) |
| 45°, 45° | √2 | √2 | 2sin(45°) |
| 90°, -90° | 0 | 0 | 0 |
Related Formulas
Product to Sum
sin(A)sin(B) = ½[cos(A-B) - cos(A+B)]
Double Angle
sin(2A) = 2sin(A)cos(A)
Half Angle
sin(A/2) = ±√((1-cos(A))/2)
Real-World Applications
Physics
Used in wave interference and superposition calculations
Engineering
Applied in signal processing and communications
Mathematics
Essential in trigonometric simplification and integration