Shauna Toh posted an instructive animated GIF from Wikipedia of the relationship between simple harmonic motion (e.g. and object bouncing up and down on a spring, a pendulum swinging back and forth, etc.) and cyclic circular motion.

And on a page on The Sine Function on the TechnologyUK site, there’s an instructive animated GIF (near the bottom of that page) of the relationship between points on a sinusoid and points on a circle. That animation shows how the angle of the point on the circle from the origin of the circle changes as the point moves around the circumference of the circle at a constant velocity, and it shows how plotting the sine of that angle results in the familiar sinusoidal shape.

Gord Anderson has a similar animation on his Quantblog WordPress site.