Sine Is a Shadow

Lesson 4 of 11 · 7 min · Free, no account needed

What you will learn

The height of a point turning on a circle, drawn over time, is the sine wave y = sin θ.
It is periodic: it repeats every 2π. Zeros come every π, tops every 2π.
The across shadow gives cos θ, the same wave shifted by a quarter turn, π/2.

A Ferris wheel turns at constant speed.
Sketch your height over one full turn. Start level with the hub.
Spin the wheel and record the shadow.
A lamp throws the seat's height onto the wall. The tape records it as the wheel turns.
Walk along the wave.
Drag the dot along the curve. The seat on the wheel moves with it. Three moments to find.
This shape is the sine wave.
Height over angle turned is y = sin θ, the sine wave.
It repeats every 2π, a full turn. A function that repeats like this is periodic, and 2π is its period.
The across shadow, x = cos θ, makes the same wave, a quarter turn earlier.
When are the two shadows equal?
Blue is the height (sin), yellow is across (cos). Tap the two moments in one turn where they match.
sin θ = cos θ when the seat is on the diagonal: at π/4 (45°) both are 0.71, at 5π/4 (225°) both are -0.71.
Slide the cos wave until it lies on the sine.
Drag the orange top sideways. How far is the shift?
cos(θ - π/2) = sin θ. The cos wave starts at its top; the sine reaches its top a quarter turn, π/2, later.
Games make things bob with a sine.
Tap the trace that this buoy code draws.
sin(t) goes from -1 to 1 and starts at 0. Times 2 gives -2 to 2, plus 5 gives 3 to 7, starting at 5.
Game: zero hunter.
The sine wave over two turns. Tap every zero and every top.
Sine is the height of a turning point, over time.
Before and after: your sketch next to the real wave. Drag the seat to compare.

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