Why can't $\sin(x)$ be expressed as an equation in terms of $x$?

Question

Why can't we write a simple equation where if we give the value of $x$ as input, we get the value of $\sin(x)$ as output?

By simple, I mean an equation involving just addition, division, subtraction and multiplication and exponentiation and keeping it in the realm of real numbers.

And I'm not necessarily asking for an equation, I'm asking if one does not exist, why is it so?

Even an intuitive explanation would work.

(It is my first question here so sorry for not being rigorous enough)

Answer 1

It can $sin(x)=\sum^{\infty}_{k=0}\frac{-(1)^kx^{2k+1}}{2k+1}$, for further information look at Taylor Series

Answer 2

Because $\sin(x)$ is not an algebraic function. It is instead a transcendental function.

Here there is a proof.