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

You can parameterized the motion in terms of the time T to complete one period and consider it's horizontal (or vertical) shadow at any t mod T.

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.

Another interesting (and fundamental) construction is to forget about circles and polygons entirely. Simply consider a periodic function over a bounded length L. Consider first the discrete case where the domain is divided into k parts. We want to find an orthonormal basis for all nicely behaved (smooth) periodic functions on this domain.

But there are infinitely many orthonormal basis sets for periodic functions on this domain. We are free to choose any. One choice is that adjacent values do not have large adjacent differences. This can be measured by squared adjacent differences. We choose that basis set that minimizes this quantity.

For the discrete case we recover DFT basis and taking limits carefully we end up with sinusoids.

\Pi will show up because of the requirement of orthonormality.