Product-to-Sum Formulas
Reference for product-to-sum trig identities for sin(A)cos(B), cos(A)cos(B), and sin(A)sin(B) with sum-to-product formulas and worked examples.
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
Limitations
- These identities apply only to products of sine and cosine — not to tangent, secant, cosecant, or cotangent
- The conversion produces a sum, which simplifies integration but can complicate direct numerical evaluation
- Not useful for finding exact angle values — use addition formulas or special angle tables for that purpose
- cos A × cos B and sin A × sin B both produce cosine terms in the result — only sin A × cos B produces sine terms; this surprises students who expect sin × sin to yield a sin result
Key Notes
- Core formulas: 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)]. These convert products of trig functions into sums or differences.
- Derived from angle addition formulas: Adding sin(A+B) + sin(A−B) = 2 sin A cos B, then dividing by 2, gives the first identity. All three formulas follow the same derivation pattern from the standard addition formulas.
- Simplifies integration: Integrals of the form ∫sin(mx)cos(nx)dx are difficult directly but become trivial after applying the product-to-sum identity — they reduce to integrals of single sinusoidal terms.
- Inverse of sum-to-product: Product-to-sum and sum-to-product are two directions of the same transformation. Choose the form based on whether you start with a product (→ sum) or a sum (→ product).
- Applications in physics: Amplitude modulation (AM radio): a carrier wave cos(ωct) multiplied by a signal cos(ωst) produces two frequency components at (ωc + ωs) and (ωc − ωs). The product-to-sum formula describes exactly how the sidebands are created.