Answer
$$D = \left\{ {\left( {x,y} \right):y \ne 0} \right\}$$
Work Step by Step
$$\eqalign{
& f\left( {x,y} \right) = \sin \frac{x}{y} \cr
& {\text{The function }}\sin \theta {\text{ is defined for all real numbers }}\theta {\text{. But }}\frac{x}{y}{\text{ }} \cr
& {\text{is not defined for the denominator 0}}{\text{. Then }}y \ne 0 \cr
& {\text{Therefore, the domain of }}f{\text{ is }}D = \left\{ {\left( {x,y} \right):y \ne 0} \right\} \cr} $$
