Product-to-sum formulas: Foundations — Maths deep dive

Why this page exists

The parent note throws around $sin A$ , $cos B$ , $A + B$ , $\frac{1}{2}$ , degrees like $7 5^{\circ}$ , and phrases like "integrate term-by-term". If any of those are fuzzy, the derivation collapses. So we list every symbol it uses or silently assumes, and for each we give:

Plain words — what it means to a human.
The picture — what it looks like on paper.
Why the topic needs it — the job it does in product-to-sum.

We order them so each one only uses things already built.

1 — An angle, and what a degree is

Picture. Stand at the centre of a clock. Point one arm straight right — that is our starting line, angle $0$ . Now sweep the other arm anticlockwise. The gap you open up is the angle.

A degree is just a unit for that gap: one full turn around the circle is split into $360$ equal slices, and each slice is $1^{\circ}$ (read "one degree"). The little raised circle $^{\circ}$ is the "degrees" mark.

Why the topic needs it. Every symbol $A$ and $B$ in the formulas is an angle. When example 3 writes $sin 7 5^{\circ}$ , the $7 5^{\circ}$ is "turn three-quarters of the way from $0$ to a right angle plus a bit". Without a picture of turning, $A + B$ and $A - B$ are meaningless.

2 — Radians: the other way to measure the turn

Picture. Lay a piece of string of length $1$ (the same length as the radius) along the curved edge of the unit circle. The angle that string subtends at the centre is $1$ radian ( $\approx 5 7^{\circ}$ ). Since the whole rim of a radius- $1$ circle has length $2 π$ , a full turn is $2 π$ radians, a half turn is $π$ , a quarter turn (a right angle) is $\frac{π}{2}$ .

Why the topic needs it. The friendly numbers $7 5^{\circ}, 6 0^{\circ}$ are easy to read in degrees, but the moment we integrate (section 8), the formula $\int cos (k x) d x = \frac{s i n ( k x )}{k} + C$ is only true when the angle is measured in radians — that clean formula silently assumes it. So the parent note lives in degrees for exact values and in radians for calculus, and you must know both are the same turning, just different rulers.

3 — The unit circle: the home of every angle

Picture. A perfectly round track of radius $1$ . An angle $A$ is a position on this track — walk anticlockwise from the far-right point by angle $A$ and stop. That stopping point is what carries all the information.

Why the topic needs it. This single picture defines $sin$ and $cos$ (next section), and it makes "adding angles" mean "turn a bit more" — which is exactly what $A + B$ is.

4 — $cos$ and $sin$ : the shadow of a spinning point

Picture. Shine a light straight down: the point's shadow on the floor slides left–right — that shadow's position is $cos A$ . Shine a light from the side: the shadow on the wall slides up–down — that is $sin A$ .

Why the topic needs it. $sin A cos B$ , $cos A cos B$ , etc. are literally these shadow-lengths multiplied together. The whole chapter is about what happens when you multiply two such shadows.

5 — The wave: unrolling the circle into a wiggle

Picture. Imagine a pen taped to the spinning point, tracing on paper that slides sideways. The circle "unrolls" into a repeating hill-and-valley curve. $cos A$ makes the same curve, just started a quarter-turn earlier.

Why the topic needs it. The parent's whole story — tuning forks, beats, "two plain notes added together" — is about waves. Multiplying $cos (2 π f_{1} t)$ by another cosine is multiplying two of these wiggles. The product-to-sum formula tells you that messy product is secretly two clean wiggles added.

6 — $A + B$ and $A - B$ : turning more, turning back

Picture. On the unit circle, $A + B$ lands you further round; $A - B$ backs you up. In example 3, $A = 7 5^{\circ}$ , $B = 1 5^{\circ}$ give $A + B = 9 0^{\circ}$ (a clean quarter turn) and $A - B = 6 0^{\circ}$ (a friendly angle) — that is why those numbers were chosen.

Why the topic needs it. Every product-to-sum output is written with $A + B$ and $A - B$ inside a $sin$ or $cos$ . These are the two new "combined" angles the formula produces.

7 — The angle addition formulas (the only assumed tool)

Plain words. These say: if you know the shadows of angle $A$ and of angle $B$ separately, here is a recipe for the shadow of the combined angle $A \pm B$ . They trade one hard shadow ( $cos (A + B)$ ) for a combination of four easy ones.

Why the topic needs it. The parent note starts from these and does nothing but add/subtract them. If these four lines feel like magic, build them first from Angle Addition Formulas. This page treats them as the single prerequisite — the raw material product-to-sum is forged from.

8 — The fraction $\frac{1}{2}$ and "multiply / divide both sides"

Picture. A chocolate bar of length $2 X$ snapped in the middle: each half is $X$ .

Why the topic needs it. Every product-to-sum formula wears a $\frac{1}{2}$ . It appears because we add two addition formulas and get $2 cos A cos B$ ; to get back to the single product we halve. The parent's biggest listed mistake is forgetting exactly this $\frac{1}{2}$ .

9 — The integral sign $\int$

Definition Integration (just enough for the examples)

$\int f d x$ means "find the running total / area under the curve $f$ ". Here:

$f$ is the curve we are adding up (for us, a cosine).
$d x$ is the differential: think of it as an infinitely thin strip of width along the $x$ -axis. The symbol $\int \dots d x$ reads as "add up $f$ times each thin strip $d x$ , all the way along $x$ " — the $d x$ tells you which variable you are sweeping along.
$k$ is any fixed number (a constant) that multiplies the variable $x$ inside the cosine — it stretches or squashes the wave. In example 2, $k$ is $3$ or $5$ .
$C$ is the constant of integration: an unknown fixed number added at the end. It appears because shifting a curve up or down by any constant does not change its slope, so "running total" is only pinned down up to that additive constant.

You need only two facts the parent uses:

You can integrate a sum piece by piece: $\int (g + h) d x = \int g d x + \int h d x$ .
$\int cos (k x) d x = \frac{sin ( k x )}{k} + C$ when $x$ is measured in radians (section 2).

Why the topic needs it. Example 2 turns $cos 4 x cos x$ (a product you cannot split) into $\frac{1}{2} [cos 3 x + cos 5 x]$ (a sum you can split), then integrates each cosine. The whole point of the chapter — "products can't be integrated term-by-term, but sums can" — lives here. Deeper study: Integration of Trig Functions.

Prerequisite map

Equipment checklist

Test yourself — cover the right side. If any answer surprises you, reread that section before opening the parent note.

What is an angle, in one phrase?

The amount of turning between two rays sharing a vertex.

How many degrees are in one full turn?

36 0^{\circ}

How many radians are in one full turn, and why?

2 π

, because that is the length of the rim of a radius-

1

circle.

Convert

9 0^{\circ}

to radians.

\frac{π}{2}

What is the radius of the unit circle?

Exactly

1

If you land at angle

A

on the unit circle, what are the point's coordinates?

(cos A, sin A)

— horizontal is

cos A

, vertical is

sin A

What is the largest value

sin A

can ever take, and why?

1

, because the point never leaves a radius-

1

circle.

What does

cos A

measure as a picture?

The horizontal (left–right) position / shadow of the point at angle

A

cos (2 π f_{1} t)

, what do

f_{1}

and

t

mean?

f_{1}

= frequency (cycles per second);

t

= time in seconds;

2 π

converts cycles to radians.

What does

A - B

mean as a motion?

Turn by

A

, then turn back by

B

— a smaller net turn.

Name the single tool product-to-sum is derived from.

The four angle addition formulas.

Why does a

\frac{1}{2}

appear in every product-to-sum formula?

Adding two addition formulas gives

2 \times

the product, so you halve to isolate it.

\int cos (k x) d x

, what are

k

d x

, and

C

k

= fixed multiplier of

x

;

d x

= the thin strip / variable you sweep along;

C

= unknown constant of integration.

What is

\int cos (k x) d x

, and what must

x

be measured in?

\frac{sin ( k x )}{k} + C

, with

x

in radians.

Can you integrate a sum term-by-term? A product?

A sum yes; a raw product no — which is exactly why we convert products to sums.

Connections

Product-to-Sum Formulas — the parent topic every symbol here feeds into.
Angle Addition Formulas — the one assumed tool; build these first if shaky.
Double Angle Formulas — the special case $A = B$ , uses the same addition formulas.
Integration of Trig Functions — where the $\int$ background is developed fully.
Wave Interference and Beats — the physical picture behind section 5's waves.