There is no analytic solution for x. Your best bet is to use Newton's method or any other approximation method that will give you as much accuracy as you desire.
Beyond an analytic solution, there isn't a unique solution. The function [itex]\mathbb R\setminus \pi\mathbb Z \to \mathbb R[/itex] taking [itex]x\to y=\dfrac{x}{\sin x}[/itex] isn't one-to-one. In fact, there are infinitely many places at which very-close-but-different values of [itex]x[/itex] are taken to the exact same [itex]y[/itex] value.
The intersection between those who would ask the OPs question and those who know who know what injective or locally injective must be null or a small number. A reasonable number that include me belong to neither class.