7.4Converting Products and Sums
Apply product-to-sum formulas (e.g. sinα cosβ = ½[sin(α+β) + sin(α−β)]) and sum-to-product formulas to rewrite and simplify trig expressions.
Product-to-sum formulas are used in calculus to integrate products of trig functions. Sum-to-product formulas appear in physics when analyzing wave interference and beats — the audible 'wah-wah' effect when two slightly different frequencies play simultaneously.
When is it useful to convert a product of trig functions into a sum, or a sum into a product — and how are these conversions derived?
The product-to-sum and sum-to-product formulas are derived by adding and subtracting pairs of sum/difference formulas. They are less commonly memorized than the core identities, but they are essential for specific calculus integration problems and for analyzing wave phenomena in physics.
How to derive product-to-sum formulas
Add the sum and difference formulas for sine:
sin(α+β) = sinα cosβ + cosα sinβ
sin(α−β) = sinα cosβ − cosα sinβ
Adding: sin(α+β) + sin(α−β) = 2sinα cosβ
Dividing by 2: sinα cosβ = ½[sin(α+β) + sin(α−β)]
Write sin(5x)cos(3x) as a sum.
Use sinα cosβ = ½[sin(α+β) + sin(α−β)] with α = 5x, β = 3x.
sin(5x)cos(3x) = ½[sin(5x+3x) + sin(5x−3x)]
= ½[sin(8x) + sin(2x)]
Write cos(4x)cos(2x) as a sum.
Use cosα cosβ = ½[cos(α−β) + cos(α+β)] with α = 4x, β = 2x.
cos(4x)cos(2x) = ½[cos(4x−2x) + cos(4x+2x)]
= ½[cos(2x) + cos(6x)]
Write sin(7x) + sin(3x) as a product.
Use sinα + sinβ = 2sin((α+β)/2)cos((α−β)/2) with α = 7x, β = 3x.
(α+β)/2 = 5x, (α−β)/2 = 2x.
sin(7x) + sin(3x) = 2sin(5x)cos(2x).
Write cos(5x) − cos(3x) as a product.
Use cosα − cosβ = −2sin((α+β)/2)sin((α−β)/2) with α = 5x, β = 3x.
(α+β)/2 = 4x, (α−β)/2 = x.
cos(5x) − cos(3x) = −2sin(4x)sin(x).
Verify: (sin(3x) + sinx)/(cos(3x) + cosx) = tan(2x)
Apply sum-to-product to numerator: sin(3x) + sinx = 2sin(2x)cos(x).
Apply sum-to-product to denominator: cos(3x) + cosx = 2cos(2x)cos(x).
Divide: 2sin(2x)cos(x) / (2cos(2x)cos(x)) = sin(2x)/cos(2x) = tan(2x). ✓
Write sin(3x)sin(x) as a sum/difference.
Hint: Use sinα sinβ = ½[cos(α−β) − cos(α+β)].
Write cos(9x) − cos(5x) as a product.
Hint: Use cosα − cosβ = −2sin((α+β)/2)sin((α−β)/2).
Write sin(6x) − sin(2x) as a product.
Hint: Use sinα − sinβ = 2cos((α+β)/2)sin((α−β)/2).
Evaluate: sin(75°) + sin(15°) using sum-to-product.
Hint: Apply sinα + sinβ = 2sin((α+β)/2)cos((α−β)/2). The result should simplify to a recognizable exact value.
Verify: (cos(3x) − cosx)/(sinx − sin(3x)) = tanx
Hint: Apply sum-to-product to both numerator and denominator, then simplify.
Interactive Practice — 5 Questions
sinα cosβ equals:
sin(5x) + sin(3x) written as a product is:
cosα cosβ written as a sum is:
cos(7x) − cos(3x) written as a product is:
Product-to-sum formulas are most useful for:
Common Mistakes
Confusing the product-to-sum formula for sinα sinβ with the one for cosα cosβ.
sinα sinβ = ½[cos(α−β) − cos(α+β)] (note the minus). cosα cosβ = ½[cos(α−β) + cos(α+β)] (note the plus).
Forgetting the negative sign in cosα − cosβ = −2sin(...)sin(...).
The cosine difference sum-to-product formula has a leading negative sign. This is the only one of the four sum-to-product formulas with a negative.
Mixing up which function goes in which position in the sum-to-product formulas.
For sinα + sinβ: the sum average goes in sine, the half-difference goes in cosine. For cosα + cosβ: both go in cosine.
Math Tips
You do not need to memorize all eight formulas. Memorize the product-to-sum formula for sinα cosβ and the sum-to-product formula for sinα + sinβ — the others follow the same pattern.
The sum-to-product formulas are the reverse of the product-to-sum formulas. If you know one set, you can derive the other by reading the equation backwards.
In physics, the sum-to-product formula for sinα + sinβ models the superposition of two waves: the 2sin((α+β)/2) factor is the carrier wave, and cos((α−β)/2) is the envelope (the beat).
When verifying an identity that involves a sum or difference of trig functions, try applying sum-to-product to both numerator and denominator — it often produces a dramatic simplification.