I think trig is assumed to be based upon triangles. This lumps trig next to squares, trapezoids, pentagons, hexagons, etc. Sure, triangles can be used to describe trig functions, but I think they do a disservice to your intuition. It's similar to Riemann sums versus integrals. True, integrals can be defined as tiny little rectangles, but I think focusing on this fact misses the bigger picture. Trigonometry is merely the relation between angles and distances. Angles and distances are the two most primal types of coordinates/measurements. Some specific styles are more popular than others, such as Cartesian and polar of course, but all of the varieties rely upon only two things: angle and distance. Trigonometry is the study of the relationship between angle and distance. Excuse my fluffy piece. I will shut up if people here don't like talking about the "why" of certain concepts. However, I think these things are important if one is to develop the highest level of intuition, and I think only the highest level of intuition is capable of breaking barriers between the status quo and the next discovery.