Numbers & Functions · 6 Trigonometric Functions

6.5 The defining feature: periodicity

What happens when the walker completes a full loop and keeps going? They retrace the exact same path. The point at angle θ + 2π is the same physical point as θ. So: sin(θ + 2π) = sin θ cos(θ + 2π) = cos θ.

The machine repeats every 2π. This is periodicity, the mathematical fingerprint of everything that cycles. Plot sin θ as the walker goes around and around, and you get the famous rolling wave.

That shape — the sine wave — is one of the most important curves in all of science. Sound is sine waves. Light is sine waves. Radio, alternating current, a plucked string, the tides — all sine waves, or sums of them.

6.6 A glimpse worth keeping

Here is the astonishing thing. Take any repeating shape at all — even a jagged square wave that jumps between two flat levels, like a light switch flicked on and off. It looks nothing like a smooth sine curve. And yet: θ the walker π 2 π 3π 2 2π θ the wave is the walker's height plotted against angle Figure 6.6. Unrolling the circle into the sine wave. The wave is the walker’s height plotted against angle; one full lap of the circle is one full period of the wave. Cosine is the same wave with a quarter-turn head start.

Every repeating signal is secretly a sum of pure sine waves — a particular sine wave, plus a faster smaller one, plus a faster smaller one still, stacked up. Add the right ones and a pile of smooth curves assembles itself into a sharp-cornered square wave.

1 wave 2 waves 3 waves many waves Stacking sine waves builds a square wave (the Fourier wonder) Figure 6.7. Building a square wave from sine waves — the Fourier teaser. One sine, then two, then three, then many: the smooth curves stack into sharp corners. (The recipe needs calculus; for now, just the wonder.) The recipe for which sine waves is called a Fourier expansion, and it’s the mathematics behind essentially all modern audio and images — every MP3, every JPEG. I won’t teach you to compute it here, because finding the recipe needs integration, two books away. Hold onto the wonder: the single curve in this Trigonometric Functions chapter is the atom from which every repeating signal in the universe is built.

All 8 chapters of Numbers & Functions · Foundations · All courses