The question is the following: what are the principal reasons to define the sine, cosine, etc., of an angle, in terms of the right triangle and not, for example, in terms of an obtuse triangle?
1) Is the pythagorean theorem a good reason for using a right triangle? Why? Also, is it true that the ratios of sides of a triangle are always functions of the angle, or this is only true in right-angled ones?
2) Also, I think that, if we define the trigonometric functions in terms of ratios of the sides, we need the right triangle, because it's the only triangle for which their sides can be localized. For example, if we have an obtuse or acute triangle, we cannot determine what is the adjacent side to the angle $A$, because there are two sides who satisfies that property, but in the right-angled triangle, the hypotenuse is always bigger and hence clearly distinguishable of the adjacent side of the angle $A$. Is this a good reason to prefer the right angles triangles to define the trigonometric ratios?
Also, every triangle can be decomposed into two right angled triangles. Is this also a good reason to prefer right triangles, or also any triangle can be decomposed into an obtuse triangle?