Product to Sum Calculator

This product to sum calculator evaluates the products sin A sin B, cos A cos B and sin A cos B for two angles, and the product to sum identities express each of those products as a sum or difference of cosines or sines. Enter two angles A and B, choose degrees or radians, and the calculator returns the three product values. These identities, historically called prosthaphaeresis, turn a multiplication into an addition, which was a calculation shortcut before logarithms and is still used in signal processing, acoustics and Fourier analysis. The identities, a worked example and the assumptions are set out below.

Angle A

Angle B

Angle unit

cos A cos B

sin A cos B

The formula

The product to sum identities are: sin A sin B = ½[cos(A − B) − cos(A + B)]; cos A cos B = ½[cos(A − B) + cos(A + B)]; and sin A cos B = ½[sin(A + B) + sin(A − B)].

Worked example

For A = 30, B = 45 degrees: sin 30 sin 45 = 0.5(0.707107) = 0.353553, which equals one half of [cos 15 minus cos 75]. Enter 30 and 45 to confirm.

Frequently asked questions

What does product to sum mean?

It rewrites a product of two trig functions as a sum or difference, for example sin A sin B becomes one half of [cos(A minus B) minus cos(A plus B)].

Where are these identities used?

In signal processing, acoustics and Fourier analysis, and historically as a multiplication shortcut called prosthaphaeresis.

Is there a reverse?

Yes, the sum to product identities go the other way, turning a sum of sines or cosines into a product.

Who this calculator is for

This calculator is for students and engineers working with trig products in analysis or signal processing.

What this calculator assumes

You enter two angles in the chosen unit.
Products are evaluated in radians.
Results are rounded for display.

Formula and sources

Related calculators

Trig Calculator: all six trig functions.
Sine Calculator: sine of an angle.
Right Angle Triangle Calculator: solve a right triangle.

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