Foundations — Sum-to-product formulas

Before you can believe a formula like $\sin A+\sin B=2\sin \frac{A+B}{2}\cos \frac{A-B}{2}$, you must own every symbol inside it. This page builds them one at a time, from nothing, in the order they lean on each other. Nothing here is assumed — if the parent note used it, we define it below.

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1. The angle $\theta$ and what "measuring an angle" means

The picture: stand at the centre of a clock. One arm points to the "3". Swing a second arm upward. The gap you swept is the angle. If you swing a little, the angle is small; a quarter turn is a right angle.

Two ways to write the same turn:

• Degrees: a full turn $=360^{\circ }$. A quarter turn $=90^{\circ }$. Used in examples like $\cos 75^{\circ }$.
• Radians: a full turn $=2\pi$. A half turn $=\pi$. Used when we solve equations and get answers like $x=\frac{n\pi }{2}$.

They are two languages for one idea. $180^{\circ }=\pi$, so $90^{\circ }=\frac{\pi }{2}$.

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2. $\pi$ — the number for "half a turn around"

The picture: take the radius of a circle as a piece of string. Lay copies of it along the curved edge. It takes $\pi$ copies to reach halfway around. That is all $\pi$ means here — a way to talk about turning without degrees.

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3. $\sin$ and $\cos$ — height and shadow on the circle

This is the heart of everything. We define sine and cosine on a unit circle (a circle of radius $1$).

Look at the figure. The blue arm is the radius pointing at angle $\theta$. Drop a straight line from its tip down to the horizontal axis. The height of that line is $\sin \theta$; the length of its shadow along the bottom is $\cos \theta$.

You read values straight off this circle: $\sin 90^{\circ }=1$ (top of the circle), $\cos 0^{\circ }=1$ (far right), $\sin 180^{\circ }=0$ (back on the axis). These are the exact readings you use to sanity-check any trig identity.

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4. $\tan$ — the ratio height-over-shadow

The picture: a steep arm is nearly vertical (big height, tiny shadow) so $\tan$ is huge. A flat arm is nearly horizontal (tiny height) so $\tan$ is near $0$.

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5. $A\pm B$, half-sum and half-difference

The picture: mark $A$ and $B$ on a number line. The half-sum sits at their midpoint; the half-difference measures half the distance from that midpoint out to either one.

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6. Product $=0$ and the equals-zero trick

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7. $n\pi$, $n\in \mathbb{Z}$ — packaging infinitely many answers

The picture: because sine has period $2\pi$ and is also zero at every half-turn, $\sin \theta =0$ not once but at $\dots ,-2\pi ,-\pi ,0,\pi ,2\pi ,\dots$ — running forwards (positive turns) and backwards (negative turns). Instead of an endless list, we write $\theta =n\pi$ — one formula that spits out every solution as $n$ steps through the whole numbers, including the negatives.

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8. The wave language: frequency $f$, envelope, carrier

This vocabulary is exactly what Beats and Superposition (Waves) and Product-to-Sum Formulas build on. When $A=B$ these ideas collapse into the Double Angle Formulas.

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Prerequisite map

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Equipment checklist

Recall Self-test: can you answer each before revealing?

What does an angle actually measure? ::: The amount of turning between two rays, not a distance. Which turn direction is a positive angle? ::: Anticlockwise is positive; clockwise is negative. How do you convert degrees to radians? ::: Multiply by $\frac{\pi }{180}$. How many radians is a full turn? ::: $2\pi$, and a half turn is $\pi$. On the unit circle, what is $\cos \theta$? ::: The horizontal position (shadow) of the point at angle $\theta$. On the unit circle, what is $\sin \theta$? ::: The vertical position (height) of the point at angle $\theta$. What is the period of sine and cosine? ::: $2\pi$ — they repeat exactly after one full turn. In which quadrant are both $\sin$ and $\cos$ negative? ::: Quadrant III (180° to 270°). What is $\tan \theta$ in terms of $\sin$ and $\cos$? ::: $\tan \theta =\frac{\sin \theta }{\cos \theta }$, the steepness. What is the half-sum of $A$ and $B$? ::: $\frac{A+B}{2}$, the midpoint angle. Why does turning a sum into a product help solve equations? ::: A product is $0$ only if one factor is $0$, so you can split into simple pieces. What does $n\in \mathbb{Z}$ mean? ::: $n$ is any whole number: $\dots ,-2,-1,0,1,2,\dots$. What is $\sin 45^{\circ }$? ::: $\frac{\sqrt{2}}{2}$. What is $\cos 30^{\circ }$? ::: $\frac{\sqrt{3}}{2}$. In beats, which piece is the slow envelope? ::: The $\cos$ of the half-difference of frequencies.