Coordinates on a Circle of Radius r

Start with the unit circle, radius 1. The point at angle theta sits at (cos(theta), sin(theta)) by definition. Now stretch that circle outward so its radius becomes r instead of 1. Every point on the circle moves straight out from the center by the same factor r, because that's what scaling a circle means: all radii grow identically. So the point that used to be at (cos(theta), sin(theta)) is now at (r cos(theta), r sin(theta)). The angle theta doesn't change, since scaling from the center never rotates anything. Only the distance from the center changes, and both coordinates feel that same stretch because x and y are both just the radius times a trig ratio.

This is why x = r cos(theta) and y = r sin(theta) work for any circle centered at the origin, no matter how big r is. The cosine and sine handle the direction, pure numbers between -1 and 1 that only care about the angle, and r handles the size, stretching that direction into an actual distance. Separating 'which way' from 'how far' is the whole trick.

When the center moves off the origin to a point (h, k), the direction and size still work the same way, but now you're measuring from (h, k) instead of (0,0). So you just slide the whole answer over: x = h + r cos(theta), y = k + r sin(theta). This sets up the general circle equation and connects directly to polar coordinates, where r and theta describe a point's location the same way.