Yonatan has a basic concept of why we use 2pi as 360 degrees. If you guys remember the winding function thread a while back. That gives a thorough explanation as to what sin, cos, and radians have to do with anything. A reminder, the winding function is a circle with radius 1. Any point on the winding funtion is called (cos(a),sin(a)) where a is the angle of rotation from the x-axis going counter-clockwise. On the winding function, the angle of rotation is also equal to the arc length made by the angle of rotation, and notice that the circumference of the winding function is 2pi. So a complete rotation around the winding function is 2pi, 180 degrees is pi and etc.

Have fun!