Since people were really interested on the sine function in that previous image, I figured I’d post this one as well. It will explain things better.
This is what the sine function of the circle actually means: it’s the y-coordinate associated with the arc length, shown in blue. A circle has circumference 2·pi·r. The unit circle has radius 1, so the full unit circle has circumference (or arclength) 2·pi.
The blue arc and the blue line shown at right have exactly the same length. This is the angle in radians.
This is the geometric definition, though. A much more powerful definition is based on infinite series.
(For the polygonal trig functions, I kept the idea of an angle around the polygon, but I gave up on the idea of using the length around the polygon - its perimeter. This caused a lot of confusion about why the side of the square didn’t trace a straight line.)