ContourPlot[((x y)/((x^2)+(y^2))),{x,-1,1},{y,-1,1},ContourShading->False,Contours->Table[(1/2)Sin[2(k π/24)],{k,-6,6}],ContourLabels->True]

ContourPlot[((x y)/((x^2)+(y^2))),{x,-1,1},{y,-1,1},ContourShading->False,Contours->N[Table[(1/2)Sin[2(k π/24)],{k,-6,6}]],ContourLabels->True]

Show[ParametricPlot3D[{{r Cos[θ],r Sin[θ],-1},{r Cos[θ],r Sin[θ],(1/2)Sin[2θ]}},{r,0,1},{θ,-π/4,π/4},PlotStyle->Opacity[0.3]],Graphics3D[{Black,Thick,Line[{{0,0,-1/2},{0,0,1/2}}]}],Boxed->False]

Show[ParametricPlot3D[{{r Cos[θ],r Sin[θ],-1},{r Cos[θ],r Sin[θ],(1/2)Sin[2θ]}},{r,0,1},{θ,-π/4,π/2},PlotStyle->Opacity[0.3]],Graphics3D[{Black,Thick,Line[{{0,0,-1/2},{0,0,1/2}}]}],Boxed->False]

Show[ParametricPlot3D[{{r Cos[θ],r Sin[θ],-1},{r Cos[θ],r Sin[θ],(1/2)Sin[2θ]}},{r,0,1},{θ,-π/4,3π/4},PlotStyle->Opacity[0.3]],Graphics3D[{Black,Thick,Line[{{0,0,-1/2},{0,0,1/2}}]}]]

Show[ParametricPlot3D[{{r Cos[θ],r Sin[θ],-1},{r Cos[θ],r Sin[θ],(1/2)Sin[2θ]}},{r,0,1},{θ,-π/4,2π},PlotStyle->Opacity[0.3]],Graphics3D[{Black,Thick,Line[{{0,0,-1/2},{0,0,1/2}}]}]]

ContourPlot[((x y)/((x^2)+(y^2))),{x,-1,1},{y,-1,1},Contours->Table[(1/2)Sin[2(k π/24)],{k,-6,6}]]