Proof there is no algorithm to compute the intersection of a line and sinusoidal wave? There is obviously a set of situations where one lack an algorithm to compute the exact solution of an equation via symbolic manipulation only, for example x = sin(x). 
One has to resort to numerical analysis and iteration methods (for example) to estimate a value resolving the above.
My question is: is there an accepted mathematical proof or set of evidence demonstrating that it is impossible to resolve this equation via symbolic manipulation or is there a possibility that one comes with a solution through some clever trick in the future?
If such a proof exist, which mathematical concepts do I need to Google for further investigation?
 A: 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. 
A: 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.
