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.