Rather than sin(), cos() and motion on a circle it is fun to consider uniform speed motion along a regular polygon and its projection hor() and ver() along horizontal and vertical directions.

This is related to DFT. As one increases the number of vertices of the regular polygon we will recover sin and cos in the limit. 2 \pi will show up in the ratio of the distance covered in one period of the uniform speed motion and the extents of the projected motion.