Skip to content
ENExplore Calculators

Product To Sum Formula Calculator

Work out product to sum formula instantly with clear inputs, formula shown and shareable results.

About Product To Sum Formula Calculator

Product-to-sum identities turn a product of sines and cosines into a sum, which is what makes Fourier analysis and beat frequencies tractable: sin A sin B = ½[cos(A − B) − cos(A + B)].

Formulas

Sine × sine

sin A sin B = ½[cos(A − B) − cos(A + B)]

Cosine × cosine

cos A cos B = ½[cos(A − B) + cos(A + B)]

Sine × cosine

sin A cos B = ½[sin(A + B) + sin(A − B)]

Worked example

With the default values below, Product To Sum Formula Calculator returns sin A · sin B via ½[cos(A−B) − cos(A+B)] of 0.26200263. Change any input above and every figure here is recalculated in your browser.

Inputs used

Angle A
50 °
Angle B
20 °

Results

sin A · sin B via ½[cos(A−B) − cos(A+B)]
0.26200263
cos A · cos B
0.60402277
sin A · cos B
0.71984631
sin A × sin B computed directly
0.26200263

Step by step

  1. 1Values used
  2. 2Sine × sine
  3. 3Cosine × cosine
  4. 4Sine × cosine
  5. 5sin A · sin B via ½[cos(A−B) − cos(A+B)]= 0.26200263
  6. 6cos A · cos B= 0.60402277
  7. 7sin A · cos B= 0.71984631
  8. 8sin A × sin B computed directly= 0.26200263

Assumptions and limits

  • The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
  • In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
  • Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

Why does the direct check match?

The identity is exact, so ½[cos 30° − cos 70°] equals sin 50° × sin 20° ≈ 0.2620026 to full precision.

Where is this used?

In signal processing: multiplying two tones produces sum and difference frequencies, which is the mathematics behind amplitude modulation and audible beats.

Related Calculators

Product To Sum Formula Calculator | Calcrivo