TRIGONOMETRYTrigonometryMathematics Calculator
Π→Σ

Product to Sum Identities

Convert sinA·sinB, cosA·cosB, sinA·cosB to sums and differences. sinA·sinB=½[cos(A-B)-cos(A+B)]. Essential for Fourier analysis and signal processing.

Concept Fundamentals
½[cos(A-B)-cos(A+B)]
sin·sin
½[cos(A-B)+cos(A+B)]
cos·cos
½[sin(A+B)+sin(A-B)]
sin·cos
Orthogonality
Fourier

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sinA·sinB=½[cos(A-B)-cos(A+B)] — product of sines becomes difference of cosines. Fourier orthogonality: ∫sin(mx)sin(nx)dx=0 when m≠n — comes from product-to-sum. Product-to-sum is the inverse of sum-to-product.

Key quantities
½[cos(A-B)-cos(A+B)]
sin·sin
Key relation
½[cos(A-B)+cos(A+B)]
cos·cos
Key relation
½[sin(A+B)+sin(A-B)]
sin·cos
Key relation
Orthogonality
Fourier
Key relation

Ready to run the numbers?

Why: Product-to-sum identities convert products to sums — essential for Fourier series, integrals of products, and solving trig equations.

How: Derived from sum/difference formulas: cos(A-B)±cos(A+B) gives 2cosAcosB or -2sinAsinB. Divide by 2 to get the product-to-sum form.

sinA·sinB=½[cos(A-B)-cos(A+B)] — product of sines becomes difference of cosines.Fourier orthogonality: ∫sin(mx)sin(nx)dx=0 when m≠n — comes from product-to-sum.
Sources:Wolfram MathWorld — Product-to-SumKhan Academy — Trig Identities

Run the calculator when you are ready.

Start CalculatingEnter angles A and B to convert products to sums

Examples — Click to Load

product-to-sum.sh
CALCULATED
$ product-to-sum --angles 30,45 --formula ext{sinSin}
Product
0
Term 1 (½)
0
Term 2 (½)
0
Sum
0
sin(A)
0.5
cos(A)
0.8660254
sin(B)
0.70710678
Identity
✓
Share:
Product to Sum Calculator Result
= 0
Identity: verified
numbervibe.com/calculators/mathematics/trigonometry/product-to-sum-calculator

Product vs Sum Components

Product vs Sum Components

Sum Term Composition

Calculation Breakdown

INPUT/CONVERSION
Angle A
30°
Angle B
45°
sin(A), cos(A)
0.5, 0.8660254
sin(B), cos(B)
0.70710678, 0.70710678
FORMULA APPLICATION
Formula
ext{Product}- ext{to}- ext{sum} ext{identity}
Product (direct)
0
PRIMARY RESULTS
Sum form result
0
IDENTITY VERIFICATION
Product = Sum
✓ Match
ext{Identity} ext{verification}

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

Π→Σ

Product-to-sum converts product of trig functions to sum/difference form.

— Wolfram MathWorld

〰️

Fourier analysis uses product-to-sum for orthogonality of sin and cos.

— MIT OCW

Key Takeaways

  • • sin(A)·sin(B) = ½[cos(A−B) − cos(A+B)] — product of sines becomes difference of cosines; note the minus
  • • cos(A)·cos(B) = ½[cos(A−B) + cos(A+B)] — product of cosines becomes sum of cosines; note the plus
  • • sin(A)·cos(B) = ½[sin(A+B) + sin(A−B)] — mixed product becomes sum of sines
  • • cos(A)·sin(B) = ½[sin(A+B) − sin(A−B)] — same structure, minus between terms
  • • Essential in Fourier analysis, signal processing, and integrating products like ∫sin(mx)sin(nx)dx

Did You Know?

📡AM radio uses product-to-sum: multiplying carrier cos(ωc t) by signal cos(ωm t) produces cos(ωc+ωm)t and cos(ωc−ωm)t — sidebandsSource: IEEE Signal Processing
∫∫sin(mx)sin(nx)dx = 0 when m≠n, π when m=n (over 0 to 2π) — orthogonality of Fourier basis comes from product-to-sumSource: Paul's Notes
🌊When two waves of frequencies f₁ and f₂ multiply, sum and difference frequencies f₁+f₂ and |f₁−f₂| emerge — the basis of heterodyningSource: MIT OCW
📐Product-to-sum formulas derive from adding/subtracting the sum and difference formulas: cos(A−B) ± cos(A+B) gives 2cosAcosB or −2sinAsinBSource: Wolfram MathWorld
🎵Audio mixers use product-to-sum when combining signals — sum and difference frequencies create new harmonicsSource: Khan Academy
🔬NIST uses these in precision frequency standards and optical comb generationSource: NIST DLMF

How Product-to-Sum Formulas Work

Product-to-sum identities convert products of trig functions into sums. They are derived by adding or subtracting the sum and difference formulas for cosine and sine.

Derivation for sin(A)·sin(B)

cos(A−B) − cos(A+B) = (cosAcosB+sinAsinB) − (cosAcosB−sinAsinB) = 2sinAsinB. So sinAsinB = ½[cos(A−B) − cos(A+B)].

Derivation for cos(A)·cos(B)

cos(A−B) + cos(A+B) = 2cosAcosB. So cosAcosB = ½[cos(A−B) + cos(A+B)].

Fourier & Integration

∫sin(mx)sin(nx)dx: use product-to-sum to get ½∫[cos((m−n)x) − cos((m+n)x)]dx. When m≠n both terms integrate to 0 over [0,2π]; when m=n, cos(0) gives π. This proves Fourier orthogonality.

Expert Tips

Never Forget the ½

Every product-to-sum formula has a factor of ½. Forgetting it doubles your result. The Sum & Difference Calculator gives the building blocks.

sin·sin and cos·cos → cos

Both use cos(A−B) and cos(A+B). sin·sin has minus between terms; cos·cos has plus. See Double Angle for the A=B case.

sin·cos and cos·sin → sin

Both use sin(A+B) and sin(A−B). sin·cos has plus; cos·sin has minus. Order matters: sin(A)cos(B) ≠ cos(A)sin(B) in the formula structure.

Integration Shortcut

∫sin(mx)sin(nx)dx over [0,2π] is 0 for m≠n, π for m=n. Same for cos. Essential for Fourier coefficients. Try the Trig Identities Calculator.

Product-to-Sum vs Other Methods

FeatureThis CalculatorDirect ProductManual Formula
All 4 product types✅✅✅
Sum form conversion✅❌✅
Identity verification✅❌⚠️ Manual
Component breakdown✅❌✅
Charts & visualization✅❌❌
Fourier context✅❌⚠️
Degrees and radians✅✅✅
7 preset examples✅❌❌

Frequently Asked Questions

How does product-to-sum connect to Fourier analysis?

When multiplying waves of frequencies f₁ and f₂, product-to-sum reveals sum and difference frequencies. Fourier coefficients involve integrals like ∫sin(mx)sin(nx)dx; product-to-sum converts these to integrable form. Orthogonality (integral = 0 for m≠n) follows directly.

Can I use these for integration?

Yes. ∫sin(mx)sin(nx)dx, ∫cos(mx)cos(nx)dx, ∫sin(mx)cos(nx)dx all become straightforward after product-to-sum. Over [0,2π], ∫sin(mx)sin(nx)dx = 0 for m≠n and π for m=n.

Why is there a ½ in every formula?

When we add cos(A−B) and cos(A+B), we get 2cosAcosB. So cosAcosB = ½[cos(A−B)+cos(A+B)]. The ½ comes from solving for the product.

What is the difference between sin·sin and cos·cos?

sin(A)sin(B) = ½[cos(A−B) − cos(A+B)] has a minus. cos(A)cos(B) = ½[cos(A−B) + cos(A+B)] has a plus. Both use cos(A−B) and cos(A+B), but the sign between them differs.

Where is product-to-sum used in signal processing?

Amplitude modulation (AM), frequency mixing, heterodyning, and demodulation. Multiplying two sinusoids produces sum and difference frequencies — the basis of radio and radar.

How do I derive sin(A)cos(B)?

sin(A+B) + sin(A−B) = (sinAcosB+cosAsinB) + (sinAcosB−cosAsinB) = 2sinAcosB. So sinAcosB = ½[sin(A+B) + sin(A−B)].

What is sum-to-product?

The reverse: sinA + sinB = 2sin((A+B)/2)cos((A−B)/2), etc. Product-to-sum converts products to sums; sum-to-product converts sums to products.

Why are these called prosthaphaeresis formulas?

Historically, before logarithms, astronomers used these to convert products to sums for easier computation — "prosthaphaeresis" means addition and subtraction.

Product-to-Sum by the Numbers

4
Product Types
½
Factor in Each
2
Terms in Sum
∫
Fourier Use

Disclaimer: This calculator provides results based on standard IEEE 754 floating-point arithmetic. Results are accurate to approximately 15 significant digits. For mission-critical applications (aerospace, medical devices), always verify with certified computational tools. Not a substitute for professional engineering analysis.

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