Sine and Cosine on the Unit Circle

Picture a point starting at (1, 0) and walking counterclockwise around a circle of radius 1 centered at the origin. As it travels, it sweeps out an angle theta measured from the positive x-axis. Wherever it lands, its coordinates are (cos(theta), sin(theta)) — that's the whole definition. Cosine tells you how far left or right the point is; sine tells you how far up or down it is. No triangle required, no degrees capped at 90.

The mechanism is just geometry: draw the terminal ray from the origin out to the circle. The ray hits the circle at exactly one point, and that point's x and y coordinates are handed to you as cosine and sine of theta. Because the circle has radius 1, the Pythagorean theorem on that point gives cos²(theta) + sin²(theta) = 1 for free. Angles can be negative, can exceed 2pi, can wrap around multiple times — the point just keeps circling, and sine and cosine keep reading off its position.

This is why sine and cosine turn into waves once you plot them against theta instead of against a circle: as the point goes around and around, its height oscillates up and down smoothly, tracing the sine curve, while its horizontal position traces the cosine curve. That unwrapping is the bridge from this topic into graphing trigonometric functions. It also sets up transformations later in the unit — stretching, sliding, and reflecting these waves all start from this same circling point.