Interpretations of $\sin(x)$ As sin(x) has an infinite number of maximum and minimums, I wondered if $\sin(x)$ could be interpreted in such a way as:
$$ax^\infty+bx^{\infty-1}\cdots zx $$Or something. Am I talking nonsense here or is there actually an interpretation of sine that involves polynomials?
 A: You're close, though what you wrote is kind of nonsense because there's no such thing as having $x^{\infty}$.
The Taylor expansion of sine is
$\sin(x)=\displaystyle\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}$, so sine is actually given by an infinite polynomial which is equal to it everywhere. You can prove this using calculus.
A: Really good question!
You would be right on that there exists such an interpretation for sine. It is called sin's Maclaurin representation and looks like this:
$$\sin x=x- \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \frac{x^9}{9!}+\dots$$
A: There are many ways of unterstanding $\sin(x)$ for example the one as a taylor series 
$$\sin(x)=\sum_{k=0}^\infty (-1)^k \frac{x^{2k+1}}{(2k+1)!}$$ 
or as 
$$\sin(x)= \frac{e^{ix} - e^{-ix}}{2i}$$
Those gives you the geometric interpretation as you are going in the complex plane and seeing the so called euler formula:
$$e^{ix} = \cos(x) + i \sin(x)$$
Well both of them really don't point that stuff out with the infinit many maximums, there are other definition like 
$$\sin(x)=x \prod_{k=1}^\infty \left( 1- \frac{x^2}{k^2 \pi^2}\right)$$
which gives you an idea that the $\sin(x)$ function really infinitly many zeroes and this one can be interpreted really nice as a polynomial (written as a product of it's zeroes).
Another way to define the $\sin(x)$ function is over a ordinary differential equation: We call $\sin(x)$ the solution of 
$$\begin{cases}
y''=-y\\
y'(0)=1\\
y(0)=0
\end{cases}$$
Here one can "see" that this functions has infinitly many zeroes, when we start at $0$ we know the function is increasing (as $f'>0$) so the function will increase, but the second derivative is lower than $0$ when $x$ is a bit bigger than $0$ so the function is increasing slower till the maximum, than it is decreasing, it will become smaller $0$ and then the idea goes on like that.
