Product-to-Sum & Sum-to-Product

Section 7.4 — converting between products and sums of trig functions

MAT 172 — Week 4
These formulas work in both directions. Product-to-Sum takes something like sin α cos β and rewrites it as a sum or difference. Sum-to-Product goes the other way — sin α + sin β becomes a product. The goal is always simplification or evaluation using special angles.
Two directions, seven formulas. Product-to-Sum converts a product like sin α sin β into a sum/difference of cosines. Sum-to-Product converts a sum like sin α + sin β into a product. Both directions are in your notes.
Product-to-Sum
sin α · sin β = ½[cos(α − β) − cos(α + β)]
cos α · cos β = ½[cos(α − β) + cos(α + β)]
sin α · cos β = ½[sin(α + β) + sin(α − β)]
Sum-to-Product
sin α + sin β = 2 sin((α+β)/2) cos((α−β)/2)
sin α − sin β = 2 sin((α−β)/2) cos((α+β)/2)
cos α + cos β = 2 cos((α+β)/2) cos((α−β)/2)
cos α − cos β = −2 sin((α+β)/2) sin((α−β)/2)
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How to remember which formula to use
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Product-to-Sum: Look at what you have. Two sines → cos difference minus cos sum. Two cosines → cos difference plus cos sum. Sin times cos → sin sum plus sin difference. Always multiply by ½.

Sum-to-Product: Look at what you have. Sum or difference of sines → 2 sin · cos with averaged angles. Sum of cosines → 2 cos · cos. Difference of cosines → −2 sin · sin (that negative sign is the one everyone drops).

💡 The key pattern: (α+β)/2 and (α−β)/2 are always the averaged angles. You're splitting the sum/difference in half for each slot.
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cos α − cos β has a negative sign — don't drop it
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cos α − cos β = −2 sin((α+β)/2) sin((α−β)/2)

Every other Sum-to-Product formula starts with +2. This one starts with −2. It's the only one. If you forget it you'll get the sign wrong on the final answer. From your notes Example 5: cos(15°) − cos(75°) = −2 sin(45°)sin(−30°). The negative is structural — it comes from the formula, not from the values.
Product-to-Sum: take a product of two trig functions and rewrite as a sum or difference. Identify which formula applies, plug in α and β, simplify.
Example 1a: 2cos(7x/2)cos(3x/2) → from your notes
1
Matches cos·cos formula: cos α cos β = ½[cos(α−β) + cos(α+β)]. But we have 2·cos·cos so the 2 cancels the ½.
2
α = 7x/2, β = 3x/2. α−β = 4x/2 = 2x. α+β = 10x/2 = 5x.
3
= 2 · ½[cos(2x) + cos(5x)] = cos(2x) + cos(5x)
2cos(7x/2)cos(3x/2) = cos(2x) + cos(5x)
Example 1b: sin(4θ)cos(2θ)
1
Matches sin·cos formula: sin α cos β = ½[sin(α+β) + sin(α−β)]
2
α = 4θ, β = 2θ. α+β = 6θ. α−β = 2θ.
3
= ½[sin(6θ) + sin(2θ)]
sin(4θ)cos(2θ) = ½[sin(6θ) + sin(2θ)]
Example 2 (exact value): cos(165°)cos(15°)
1
cos·cos formula: ½[cos(α−β) + cos(α+β)]. α = 165°, β = 15°.
2
α−β = 150°, α+β = 180°.
3
= ½[cos(150°) + cos(180°)] = ½[−√3/2 + (−1)]
4
= ½[−√3/2 − 1] = −√3/4 − ½ = (−√3 − 2)/4
cos(165°)cos(15°) = (−√3 − 2)/4
Example 3: 2cos(23°)sin(17°) — note the order: cos·sin
1
Rewrite as sin·cos: sin(17°)cos(23°). Use sin·cos formula: ½[sin(α+β) + sin(α−β)]
2
α = 17°, β = 23°. But we have 2·sin·cos so factor: 2·½[sin(40°) + sin(−6°)]
3
= sin(40°) + sin(−6°). Apply odd property: sin(−6°) = −sin(6°)
2cos(23°)sin(17°) = sin(40°) − sin(6°)
Sum-to-Product: take a sum or difference of two trig functions and rewrite as a product. The averaged angles (α+β)/2 and (α−β)/2 always go in the slots.
Example 4a: sin(4θ) − sin(2θ)
1
sin α − sin β = 2sin((α−β)/2)cos((α+β)/2). α = 4θ, β = 2θ.
2
(α−β)/2 = (4θ−2θ)/2 = θ. (α+β)/2 = (4θ+2θ)/2 = 3θ.
3
= 2sin(θ)cos(3θ)
sin(4θ) − sin(2θ) = 2sin(θ)cos(3θ)
Example 4b: cos(7x) + cos(−7x)
1
Use even property first: cos(−7x) = cos(7x). So this is cos(7x) + cos(7x) = 2cos(7x).
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OR use Sum-to-Product: α = 7x, β = −7x. (α+β)/2 = 0. (α−β)/2 = 7x.
3
= 2cos(0)cos(7x) = 2·1·cos(7x)
cos(7x) + cos(−7x) = 2cos(7x)
Example 5 (exact value): cos(15°) − cos(75°)
1
cos α − cos β = −2sin((α+β)/2)sin((α−β)/2). α = 15°, β = 75°. Note the negative!
2
(α+β)/2 = 45°. (α−β)/2 = −30°.
3
= −2sin(45°)sin(−30°). Apply odd: sin(−30°) = −sin(30°) = −½.
4
= −2(√2/2)(−½) = −2·(−√2/4) = √2/2
cos(15°) − cos(75°) = √2/2
Example 7: Simplify [cos(4t) − cos(2t)] / [sin(4t) + sin(2t)]
1
Numerator: cos α − cos β = −2sin((4t+2t)/2)sin((4t−2t)/2) = −2sin(3t)sin(t)
2
Denominator: sin α + sin β = 2sin((4t+2t)/2)cos((4t−2t)/2) = 2sin(3t)cos(t)
3
= −2sin(3t)sin(t) / 2sin(3t)cos(t). Cancel −2sin(3t) / 2sin(3t) = −1.
4
= −sin(t)/cos(t) = −tan(t)
[cos(4t) − cos(2t)] / [sin(4t) + sin(2t)] = −tan(t)
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All 7 formulas

Product → Sum
sinα sinβ = ½[cos(α−β)−cos(α+β)]
cosα cosβ = ½[cos(α−β)+cos(α+β)]
sinα cosβ = ½[sin(α+β)+sin(α−β)]
Sum → Product
sinα+sinβ = 2sin((α+β)/2)cos((α−β)/2)
sinα−sinβ = 2sin((α−β)/2)cos((α+β)/2)
cosα+cosβ = 2cos((α+β)/2)cos((α−β)/2)
cosα−cosβ = −2sin((α+β)/2)sin((α−β)/2)

Exam reminders

cosα−cosβ starts with −2 not +2. Only one with a negative.
Product-to-Sum always has ½ out front.
Sum-to-Product uses averaged angles: (α+β)/2 and (α−β)/2.
sin is odd: sin(−x) = −sin(x). cos is even: cos(−x) = cos(x). Use these to simplify negative angles.
For exact values: get it down to known angles (30°, 45°, 60°, 150°, 180°) then evaluate.
Simplify fractions: look for common factors in numerator and denominator after applying formulas.