Wave Patterns

Pre-Calculus · Axiom Academy

One spinning point, and the smooth repeating curve it leaves behind — the wave at the heart of trigonometry. A guitar string, a swinging pendulum, the tide, the voltage in the wall — they all rise and fall in the same smooth, endlessly repeating shape. That shape isn't a coincidence. It falls straight out of something as simple as a point going around a circle, and the function that captures it is the sine. Watch one point travel once around the circle on the left. Its height above the center is the only thing we track — and as the angle turns, that height gets carried to the right and drawn out into a curve. The curve it leaves behind is exactly . The sine wave is just the height of a point going around a circle — read off, moment by moment. Make it taller, make it faster Every wave you'll meet is a sine curve with two dials. Slide amplitude to set how tall it swings, and slide how fast it cycles to set how tightly it repeats. The curve below is always the true plot of — watch the height and the period change as you turn each dial. Same curve, two dials: amplitude sets the height, the cycle speed sets the period — how often the pattern repeats. A wave is periodic : its shape repeats forever. Drag the handle to shift the curve left and right and you'll see the exact same hump come around again and again. Whatever happens on one stretch happens again one full period — — later, with no end.

This is the written version of the interactive lesson above. See the full Pre-Calculus course.