We all know the elementary school definition of the trigonometric functions: In a right triangle, $\cos$ is defined to be the ratio of the adjacent and the hypotenuse, etc.
But, I observed that this definition has one "problem": For a given angle, there are infinitely many right triangles having that angle. How does one prove that the ratio $$\frac{\text{adjacent}}{\text{hypotenuse}}$$ is independent of the right triangle used? The same goes for the other ratios.
Proving the "opposite direction" seems easy: the angle between vectors $u$ and $v$ is $$\arccos \frac{u\cdot v}{|u||v|}$$ and if the vectors $u'$ and $v'$ are constant multiples of $u$ and $v$, respectively, then we see that the common factor cancels out nicely in the $\arccos$ formula.