Product-to-Sum and Sum-to-Product

When Two Waves Meet

Pluck two guitar strings tuned almost the same and you hear a slow “wah… wah… wah” pulsing over the note. Those pulses are called beats, and they are pure trigonometry.

The top two rows are the separate waves, each a steady sine. Add them and you get the bottom wave (green), which speeds along inside a slow envelope (yellow) and pulses loud, soft, loud, soft. Adding two sines produced a product: a fast wave times a slow one. That is the whole idea of this lesson.

Sums into Products

When you add two sines or two cosines, you can rewrite the sum as a product:

$\sin A + \sin B = 2\sin\frac{A+B}{2}\cos\frac{A-B}{2}$

$\cos A + \cos B = 2\cos\frac{A+B}{2}\cos\frac{A-B}{2}$

The first factor uses the average of the two angles. The second uses half their difference.

In the beats animation, $\frac{A+B}{2}$ is the fast wave you hear as the tone, and $\cos\frac{A-B}{2}$ is the slow envelope you hear as the pulsing. When $A$ and $B$ are close, that difference is tiny, so the envelope is very slow. That is why slightly-out-of-tune strings beat slowly.

The Differences Too

Subtraction works the same way, it just swaps a sine and a cosine:

$\sin A - \sin B = 2\cos\frac{A+B}{2}\sin\frac{A-B}{2}$

$\cos A - \cos B = -2\sin\frac{A+B}{2}\sin\frac{A-B}{2}$

Watch the minus sign in front of the cosine difference. It is easy to drop.

Products into Sums

The reverse direction is just as useful. A product of sines and cosines becomes a sum:

$\sin A \cos B = \tfrac{1}{2}\big[\sin(A+B) + \sin(A-B)\big]$

$\cos A \cos B = \tfrac{1}{2}\big[\cos(A-B) + \cos(A+B)\big]$

$\sin A \sin B = \tfrac{1}{2}\big[\cos(A-B) - \cos(A+B)\big]$

Where They Come From

Every one of these is just the addition and subtraction formulas, added or subtracted.

Line up the two sine formulas:

Add them. The $\cos A \sin B$ terms cancel:

$\sin(A+B) + \sin(A-B) = 2\sin A \cos B$

Divide by 2 and you have the product-to-sum rule for $\sin A \cos B$. The sum-to-product rules are the same identities read backwards, letting $A + B$ and $A - B$ become the new angles.

Example: Product to Sum

Rewrite $2\sin 75° \cos 15°$ as a sum.

A messy product turned into two clean, known values.

The Full Set

Sum to product:

Product to sum: