[man-and-laptop]

I just derived the spherical law of cosines that's the foundation of this formula using quaternions and Sympy. The quaternions were modelled as 4x4 matrices.

The philosophy is that you can treat trigonometry as a triangle construction problem. You give me the specs of a triangle, and I can use physical tools like compasses, protractors and rulers to construct your triangle. This is similar to using trigonometry, except without any algebra involved, just construction tools. You can also use this philosophy to derive the algebraic formulas. When it came to formula derivation, I used the quaternions to simulate the physical construction, and I got out the formulas.

[theoh]

The last figure on this page http://www.vias.org/calculus/07_trigonometric_functions_01_0... really has that character (in 2D).

I wish trigonometry in secondary school had been taught more like this (geometrically) because it would have been good preparation for vector calculus, among other things.

[jakeogh]

That sounds like fun. Please drop a link if you post it.

[man-and-laptop]

See here: https://github.com/jkabrg/Spherical-law-of-cosines-proof-usi...

I'll provide more explanation later.