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?