Sine and cosine

For an angle $\theta$ measured in radians from the positive x-axis, the point at that angle on the unit circle is $P = (\cos\theta, \sin\theta)$: cosine is its horizontal position, sine its height. A radian is a length along the unit circle, so θ is also the arc travelled from (1, 0), and both functions repeat after one turn, a period of $2\pi$. This page follows one turn, $0 \le \theta \le 2\pi$.

The three parts share one unit of length. The height of P, carried across, traces the sine wave with θ to the right; its horizontal position, carried down, traces the cosine wave hanging below with θ downward. The yellow arc from (1, 0) to P has length θ, and the same yellow length lies along both θ axes: one turn unrolls to 2π ≈ 6.283 radii. Flat turns the hanging graph a quarter turn about the centre. That rotation takes each point of the hanging graph to height cos θ at position θ, and takes P to Q, the point a quarter turn ahead. Q's height is P's horizontal position, so the cosine wave is level with the sine wave $\pi/2$ further on: it is the sine wave shifted $\pi/2$ to the left.

Turning at one radian per second, P has velocity $(-\sin\theta, \cos\theta)$, of length 1 and at right angles to the radius. Its upward part, cos θ, is the rate at which the height changes, which is the slope of the sine wave: 1 at θ=0, −1 at θ=π, and 0 at peaks and troughs. Spin leaves a red tick every π/6 s. The ticks on the circle are evenly spaced; their shadows bunch near the top and bottom, where the height changes slowly.