Why are trigonometric functions defined in the context of right triangles? Couldn't we just use a different, fixed angle? Is there something special about the ratios of the sides in a right triangle? Is the pythagorean theorem somehow involved?
 A: Say you have two right triangles, and in addition to the right angle, they both have an angle $v$. In that case they are similar, and thus the ratio of two sides in one triangle is equal to the ratio of the two corresponding sides of the other triangle. These ratios have been calculated beforehand, for convenience, and we call them $\sin v,\cos v$ and $\tan v$. (Before calculators, you used to have to look it up in tables. You could basically go to any book store with a science section and buy a sine table.)
The reason we use right triangles is probably because that's easiest. You only get one single number to depend on (the angle $v$) as opposed to several. Right triangles are also very common in practical applications like construction, simply because right angles themselves are so common.
A: One reason I found out so far is the formula $ \sin^2(x) + \cos^2(x) = 1,$ which is derived from the Pythagorean theorem.
Edit: 
The 90° Angle is also used in the Proof of the Law of sines and, not surprisingly again by means of the Pythagorean Theorem, in the Proof of the Law of cosines.
As far as i know it's used in more trigonometric formulas, but i don't know in which and how yet.
