Track the height of a point spinning around a circle and plot it over time — it rises and falls into the smooth hills of a sine wave. That's how circular motion becomes the waves behind sound, light, and tides.
Unrolling the circle
Track the height (sine) of the circling point as the angle grows, and plot it against the angle. You get a smooth, endlessly repeating wave: up to 1, down to −1, over and over every 2π. That is the graph of y = sin x — a circle unrolled into a timeline.
Amplitude stretches the wave taller (y = A sin x); frequency packs more cycles into the same space (y = sin Bx).
- Amplitude A: half the height from trough to crest — loudness, brightness, swing size.
- Period 2π/B: the length of one full cycle before the wave repeats.
- Frequency: cycles per unit — the reciprocal of the period. Higher pitch, higher frequency.
Sound, light, tides, alternating current, your heartbeat on an ECG — anything that cycles is built from sine waves. Trigonometry is the alphabet of vibration.
- Sketch one period of y = sin x, marking where it crosses zero and where it peaks.
- On the same axes, sketch y = 2 sin x. What changed — the height or the spacing?
- Now sketch y = sin 2x. What changed this time?
- Label the amplitude and period of all three curves.
What you should see: You saw that the coefficient outside sine controls amplitude while the one inside controls period — two independent knobs on every wave.