Interesting. versine has a lovely and intuitive geometric definition. If you construct a right triangle from the origin to some point on the circle, most people who have done trig will know that the x-coordinate of that point is r cos theta, where theta is the angle and r is the radius. Geometrically the distance from the origin to where the triangle rests on the x axis is r cos theta. But what about the rest of that radius? ie the line segment on the x-axis from there to where the circle intersects the x-axis?

That is r versin theta (ie r - r cos theta). Pretty cool no? I mean I've literally never had to find the length of that line, but that's how you would if you wanted to..