Sinusoidal Modeling from Data

Lots of real quantities rise and fall in a repeating rhythm because they're tied to something going around in a circle: the Earth spinning and tilting, tides pulled by the Moon's orbit. Daylight hours trace a wave across the year because the Earth's tilt carries a city from more direct sunlight to less and back, once per orbit. Any time a quantity oscillates smoothly between a high and low value on a regular cycle, a sine or cosine function can model it.

To build the model, you pull four numbers out of the data. The amplitude is half the distance between the max and min values. The midline is the average of the max and min, which tells you where the wave centers vertically. The period is the time for one full cycle, found by doubling the time between a max and the next min, or measuring max to max. The phase shift tells you where in the cycle the data starts, and it decides whether cosine or sine, and which direction, fits most naturally — cosine is the easy choice when you're given a peak or trough time to start from.

This is where the abstract sine and cosine graphs from earlier in the unit meet messy, real numbers: tide charts, sunrise tables, temperature logs. The same stretches and shifts that transform y = cos(x) into A·cos(B(x-C))+D are exactly the four quantities you extract from data. The catch is that a model built from one year's data assumes the pattern repeats exactly, so it gets shakier the farther you project it, since real tides and weather drift from true periodicity over time.