Sine and Cosine

The wave behind sound, light and the logo on this page: amplitude, frequency, phase and midline, one parameter at a time.

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video6:05 · 7 chapters

Sine wave: a simple explanation on a Ferris wheel

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step 01 / 05 · A wave from one rule

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A wave from one rule
The sine function turns an angle into a height. Plotted against the angle it makes a smooth wave that repeats every $2\pi \approx 6.28$ .

It peaks at $x=\pi/2$ , where $\sin x = 1$ .
Amplitude: how tall
Multiply by $A$ to stretch the wave vertically:
$y = A\sin x$
With $A=3$ the wave swings between $-3$ and $3$ . The dashed curve is plain $\sin x$ .
Frequency: how fast it repeats
Put a multiplier inside: $y=\sin(\omega x)$ . With $\omega = 2$ the wave runs twice as fast, so one full cycle takes
$T = \frac{2\pi}{\omega} = \pi$
instead of $2\pi$ .
Phase: where it starts
Adding a constant inside, $y=\sin(x+\varphi)$ , slides the wave sideways. With $\varphi=\pi/2$ the wave starts at its peak, which is exactly the cosine function:
$\sin\left(x + \tfrac{\pi}{2}\right) = \cos x$
Lifting the midline
Finally, add a constant outside: $y = \sin x + d$ . With $d=2$ the wave oscillates around $y=2$ instead of $y=0$ .

Together, $y = A\sin(\omega x + \varphi) + d$ describes any simple wave: height, speed, start and centre.

A wave from one rule