Modeling things that repeat
A wheel turns, a tide comes in, a spring bobs. How do a few measured facts about a steady cycle become a formula that predicts it?
Where you are 3 inside
What this area is about
Many quantities rise and fall on a steady cycle: a seat on a turning wheel, the water at a harbor, a weight on a spring. A sine or cosine describes each, and every number in the formula is a fact about the situation: amplitude, midline, period and where the cycle stands when the clock starts. With the input in radians, a period P means \displaystyle b = \frac{2\pi}{P}.
How these fit together
All three apply one model, y = A\cos\bigl(b(t - c)\bigr) + k or its sine form, and differ in where the numbers come from.
Height on a turning wheel reads them off the wheel: radius, hub height, time per turn, starting position. Tides, daylight and sound builds them from facts in words: highest and lowest values, period and a peak time. Simple harmonic motion is the concept behind a spring, or a pendulum on small swings, d(t) = a\cos(\omega t) or d(t) = a\sin(\omega t) with \omega in radians per second, read as the height of a point turning on a circle.
A path through it
No lesson here is a prerequisite of another, but they point to each other. The wheel is the most concrete and the other two refer back to it, so start there; tides and simple harmonic motion follow.
The wheel and tide lessons ask a model for the value at a given time and for the times it reaches a given level. The second needs inverse trigonometry, whose answer lies in a restricted range, so the other times come from symmetry and from adding whole periods.