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In a numerical methods class I'm taking, it was claimed that the equation $A = \frac{R^2}{2} \left(\theta - \sin\theta \right)$ cannot be analytically solved for $\theta$. I don't doubt that this is true, but I'm curious how it could be proved that this is true.
In general, how would one go about proving that a solution to an equation in terms of elementary functions does not exist?
About a century and a half ago, Liouville developed a criterion to determine whether the solution to certain types of differential equation could be expressed in elementary terms. This criterion was powerful enough to show for example that there is no elementary function whose derivative is $e^{-x^2}$.
The ideas of Liouville led to the field of differential algebra.
Much later, Risch produced an algorithm that will determine, for a fairly wide class of elementary functions, whether a function has an elementary antiderivative. There have been improvements since, and improved algorithms have been at least partly implemented.
Mainly, Liouville and Risch treated integration in finite terms and answered, with their method, the problem of integration of elementary functions by elementary functions.
For the problem of solving equations in terms of elementary functions, Liouville's representation of elementary functions as composition of explicit algebraic functions and elementary standard functions, or, as composition of explicit algebraic functions, $\exp$ and $\ln$, is necessary.
Solving an equation in terms of elementary functions means that the function which is set constant by this equation has an inverse function and that this inverse function is expressable in terms of elementary functions.
Assume that your equation exists: A and R, not dependent on $\theta$, are choosen so that the equation depends on its solving variable ($\theta$).
Write your equation as zeroing equation. Then look at the structure of this equation. The solving variable ($\theta$) is argument of algebraic component functions (The identity function ($\theta\mapsto \theta$) is an algebraic function.) and of transcendental component functions ($\sin$). The component functions are the arguments of an outer function which is an algebraic function. This algebraic function can be represented as a unary algebraic function only for the values of $\theta$ where the values of the component functions are algebraically dependent. An algebraic function $A$ can have an inverse function which is expressable in terms of elementary functions only if $A$ is unary. Therefore your equation can have a solution in terms of elementary functions only if a value of $\theta$ where the values of the component functions are algebraically dependent is a solution of the equation. The equation from your example can have a solution which is expressable in terms of elementary functions only when $0$ is a solution of this equation.
Look for transcendental number theory, e.g. for Schanuel's conjecture and its precursors, and for the expression of elementary functions as composition of explicit algebraic functions, $\exp$ and $\ln$.
