Proof there is no algorithm to compute the intersection of a line and sinusoidal wave?

You may be interested in Richardson's theorem, an amazing theorem which implies that the problem of determining for a given mathematical expression $E(x)$, of particularly simple form, whether $E(x)=0$ for all real $x$ is undecidable. Not only is there no computable algorithm to determine the answers to all such questions, but for any given axiomatic system such as ZFC there are some expressions $E$ for which the fact of the matter of whether $E(x)$ is identically $0$ is not settled by those axioms.

Your question is addressed in my paper What is a closed-form number?

The first step is to decide which "symbols" or functions you accept as furnishing a "symbolic solution." In my paper I focus on perhaps the most restrictive set, namely exp and log and the arithmetic operations.

The second issue you have to address is whether you're asking for a symbolic expression for a function or for a symbolic expression for a number (namely, the root of some equation that you're trying to solve). This distinction seems not to have been emphasized in the literature prior to my paper. There is quite a bit of literature about closed-form functions, but knowing, for example, that the smallest positive root of the equation $kx = \sin x$, thought of as a function of $k$, has no closed-form expression, doesn't answer the question of whether there is some ad hoc expression for specific values of $k$.

In my paper I show that Schanuel's conjecture implies that there is no closed-form expression for the roots of equations such as $x + e^x = 0$. I think that this is pretty much the best answer we can currently give to this type of question.