Unwrapping the Circle into a Sine Wave

Picture a point starting at (1, 0) on the unit circle and walking counterclockwise as the angle theta increases. At every instant that point has a height above the x-axis, and that height is sin(theta) by definition. The circle keeps the point's horizontal position bounded, but its height rises and falls as it goes around. If you track only that height and plot it against theta instead of against the point's position on the circle, you stretch the circular motion out along a straight timeline. That stretching is the unwrapping: the circle's rim becomes the horizontal axis, and the point's bobbing height becomes the curve.

Trace one full lap. At theta = 0 the point sits at height 0. By theta = π/2 it has climbed to the top of the circle, height 1. Continuing to theta = π brings it back down to height 0, then to theta = 3pi/2 it dips to the bottom, height -1, and by theta = 2pi it is back at height 0, finishing one complete wave. That is why the sine graph rises, peaks, falls through zero, bottoms out, and returns, all within one span of 2pi.

Once theta passes 2pi, the point is simply lapping the circle again, retracing the exact same heights in the exact same order. That is why sine is periodic with period 2pi: sin(theta + 2pi) = sin(theta) always. This unwrapping picture is what makes the later stretching and sliding of sine and cosine graphs make sense, since those transformations just change how fast the point circles or where it starts.