Sum-to-Product Formulas — Mathematics Knowledge System

Overview

The sum-to-product formulas rewrite a sum or difference of two sines (or two cosines) as a product of sines and cosines of half-sum and half-difference angles. They are the algebraic inverse of the product-to-sum formulas, and they are obtained directly from the angle sum and difference identities by adding or subtracting pairs of equations and substituting new variables for the half-sum and half-difference. These identities are indispensable whenever a sum of two oscillations needs to be seen as a single oscillation — the classic example is acoustic beats, where two tones of nearly equal frequency combine into one tone whose amplitude itself oscillates slowly.

Intuition

Start from the sum formulas sin(P+Q) = sinP cosQ + cosP sinQ and sin(P−Q) = sinP cosQ − cosP sinQ. Adding these two equations cancels the cosP sinQ terms and doubles sinP cosQ, giving sin(P+Q) + sin(P−Q) = 2 sinP cosQ. Renaming A = P+Q and B = P−Q (so P is the average angle and Q is the half-difference) turns this into sinA + sinB = 2 sin((A+B)/2) cos((A−B)/2) — exactly the sum-to-product identity. The same add/subtract trick applied to the cosine sum formulas produces the other two identities. Geometrically, two waves added together average into one wave at the mean frequency, modulated by a slower envelope at the half-difference frequency — which is precisely what the product form displays.

Formal Definition

Definition

For any angles A and B, writing P = (A+B)/2 and Q = (A−B)/2 so that A = P+Q and B = P−Q:

\sin A + \sin B = 2\sin\!\left(\frac{A+B}{2}\right)\cos\!\left(\frac{A-B}{2}\right)

Sum of sines

\sin A - \sin B = 2\cos\!\left(\frac{A+B}{2}\right)\sin\!\left(\frac{A-B}{2}\right)

Difference of sines

\cos A + \cos B = 2\cos\!\left(\frac{A+B}{2}\right)\cos\!\left(\frac{A-B}{2}\right)

Sum of cosines

\cos A - \cos B = -2\sin\!\left(\frac{A+B}{2}\right)\sin\!\left(\frac{A-B}{2}\right)

Difference of cosines

Worked Examples

Apply the sum-of-sines formula with A = 75°, B = 15°.
$\sin75° + \sin15° = 2\sin\!\left(\frac{75°+15°}{2}\right)\cos\!\left(\frac{75°-15°}{2}\right) = 2\sin(45°)\cos(30°)$
Substitute exact values sin45° = √2/2, cos30° = √3/2.
$= 2\cdot\frac{\sqrt2}{2}\cdot\frac{\sqrt3}{2} = \frac{\sqrt6}{2}$

Answer: sin(75°) + sin(15°) = √6/2 ≈ 1.2247

Practice Problems

Difficulty 3/10

Find the exact value of sin(40°) + sin(20°) using the sum-to-product formula.

Difficulty 4/10

Find the exact value of cos(110°) − cos(70°) using the difference-of-cosines formula.

Difficulty 6/10

Two tuning forks emit tones sin(2π·256t) and sin(2π·260t) (frequencies in Hz). Using the sum-to-product formula, identify the frequency of the audible tone and the frequency of the slower amplitude envelope (the beat frequency) heard when the tones are added.

Quiz

Summary

sinA ± sinB and cosA ± cosB can be rewritten as products of sines/cosines of the half-sum (A+B)/2 and half-difference (A-B)/2.
They follow from adding or subtracting the angle-sum and angle-difference formulas.
They are the standard tool for analyzing beat phenomena, where two nearly-equal frequencies combine into a carrier tone modulated by a slow envelope.

References

WebsiteWikipedia — List of trigonometric identities