>> What struck me as odd when I was an undergraduate, and still strikes me to this day, is that none of these are the obvious trigonometric definitions about the opposite and adjacent sides of a right triangle.

I had the less common experience of learning some graphics programming prior to taking a trig class in school. "How do I draw a circle?" You sweep an angle from 0 to 360 (or 2*pi) and use sin() and cos() to get points on the unit circle, then multiply by your radius and plot them. For me the notion of sin and cos being coordinates of points on a unit circle was natural. Later when I had trig class it seemed really weird to define these functions as ratios of sides of a triangle. In particular you never talk about a side length as being negative. So no, it's not universal that the triangle definitions seem more obvious or make more intuitive sense.

I think this is important for people to understand. What seems easy or natural can depend on how your particular tree of knowledge was constructed.