What are the product-to-sum formulae?
2 sinA cosB = sin(A+B) + sin(A−B); 2 cosA cosB = cos(A−B) + cos(A+B); and 2 sinA sinB = cos(A−B) − cos(A+B). They rewrite a product as a sum or difference.
// maths › Trigonometric Identities
Product-to-Sum Formulae
Convert a product of sines and cosines into a sum or difference (and back), with the full identity worked step by step, in degrees or radians.
2sinAcosB=sin(A+B)+sin(A−B); 2cosAcosB=cos(A−B)+cos(A+B)
Angle unit
Frequently asked questions
Can you show a worked example?
2 sin50° cos20° = sin70° + sin30° ≈ 0.9397 + 0.5 = 1.4397. In radians 50° ≈ 0.8727 and 20° ≈ 0.3491 rad; the toggle re-derives the same value in radians.
Why are they useful?
A sum is far easier to integrate than a product, so these identities are essential in calculus. The reverse, sum-to-product, factorises expressions and explains beats.
What is a beat?
When two notes of nearly equal frequency add, the sum-to-product identity shows the result as a slow amplitude wobble — the beat — riding on the average frequency.
Where are they used?
Musical beats, amplitude modulation in radio, and integrating products of sinusoids in physics and engineering.