Why do trigonometric functions work? Sine is equivalent to the opposite side over the hypotenuse. It's formula (simplified) would be this:
$$\sin x = x - \frac{x^3}{6} + \frac{x^5}{120} - \frac{x^7}{5040} + \frac{x^9}{362880} - \cdots $$
However, I really want to know: why does this work? Why can this specific function be used on any right angle, and produce the opposite over the hypotenuse? The formula itself almost seems random. How does it relate to those sides?
 A: The formula is an example of what is called a "Taylor series" which is based on the concept of derivatives. The correctness of the formula for the sine function is intimately connected to the differential connections between sine and cosine, namely, that $\sin' = \cos$ and $\cos'=-\sin$.
These two facts can be straightforwardly demonstrated using geometry, based on the usual definitions you know for sine and cosine. (If you know the angle addition formulae $\sin(a+b)=\sin a\cos b+\cos a\sin b$ and $\cos(a+b)=\cos a\cos b-\sin a\sin b$ you're already halfway there). Then all that's left is calculus and the theory of Taylor series.
Also, note that the formula might appear less random if you rewrite it as
$\sin x = \sum_{k=0}^{\infty}\frac{(-1)^k}{(2k+1)!}x^{2k+1}$
Which in turn will appear less random once you realize the general Tayolr series is
$f(x) = \sum_{k=0}^{\infty}\frac{f^{(k)}(0)}{k!}x^k$
So really the only thing left to show is that the repeated derivatives of sine are $0, 1, 0, -1$ repeating in a cycle.
