Product-to-Sum and Sum-to-Product Calculator

Two identities that swap products and sums

Trigonometry has a matched pair of identity families. Product-to-sum turns a product of two sines or cosines into a sum, and sum-to-product runs the other way, turning a sum into a product. This calculator does both: pick the expression you have, enter the two angles, and it returns the converted form and its value.

Product to sum

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

These are the workhorses of integration. An integral like ∫ sin(3x)cos(x) dx looks nasty until you rewrite the product as ½[sin(4x) + sin(2x)], which then integrates term by term in seconds.

Sum to product

sin A + sin B = 2 sin((A+B)/2) cos((A−B)/2) sin A − sin B = 2 cos((A+B)/2) sin((A−B)/2) cos A + cos B = 2 cos((A+B)/2) cos((A−B)/2) cos A − cos B = −2 sin((A+B)/2) sin((A−B)/2)

These help you factor expressions, solve equations, and explain beats in acoustics. Two tones at nearby frequencies, cos(2πf₁t) + cos(2πf₂t), combine into one tone at the average frequency whose volume swells and fades at the difference frequency. That slow throb you hear when tuning a guitar against a reference note is exactly the (f₁ − f₂) term.

A surprise worth flagging

Both cos·cos and sin·sin convert to cosine terms. Only sin·cos produces sine terms in the result. Students routinely expect sin·sin to give a sine, and it does not.

Where they came from

Both families fall out of the angle addition formulas. Add sin(A+B) and sin(A−B) and the cross terms cancel, leaving 2 sin A cos B, which rearranges into the first product-to-sum rule. Long before calculators existed, astronomers used these identities, under the tongue-twisting name prosthaphaeresis, to turn slow multiplication into fast addition, the same labor-saving trick logarithms later made famous.