TITLE: Proving Trigonometric “Definitions”
QUESTION [6 upvotes]: The expression trigonometric “definitions” refers here, rather narrowly, to statements expressing stable relations between the sides of right triangle. 
Thus, for instance, the traditional definition of sine supposes that one has a demonstration that the ratios between opposite side and hypothenuses are independent of the size of the right triangle and are dependent instead on the amplitude of the angles of the right triangle.
My question is the following: How these stable relations, simply assumed by most trigonometric “definitions,” can actually be demonstrated?
A very similar question has been posted here but I remain unsatisfied by most answers — which seem to convoke either unnecessary complex mathematical objects or simplistic historical accounts. 
I guess, what I am looking for, is something like a geometrical proof — but am open to others!

REPLY [2 votes]: You should look up similarity. In particular, similarity of triangles, and especially right triangles.
It is because any two right triangles containing the same angle are similar that the functions depend only on the angle in question.
As for why right triangles. Why not, say, isosceles triangles, since these also allow one to define a certain trig function of the vertex angle uniquely, especially as they are laterally symmetric? Well, there's no reason why one couldn't do that; indeed, the first compiler of trig tables compiled them for isosceles triangles, and called them chords of the vertex angle. However, one can argue that that would not be as fundamental as using right triangles for the simple reason that whereas not all triangles can be divided into two isosceles triangles, all can always be partitioned into two right triangles. This keeps things simpler. NB. What was called the chord of $x$ is now double the sine of $x/2.$