Do you have any polar graph paper? That would make it easy to graph. Just set f(r,theta) equal to each of your z's and then solve for r in terms of theta. So for instance:
z = 1 = 1+ sin(T) - r ----> r = sin(T)
So start at theta=0 on your polar graph paper and then work your way counterclockwise calculating r along the way. Then do this for each z.
Alternatively, if you must have cartesian coordinates, remember:
r = (x^2+y^2)^(1/2)
sin(T) = y/r = y*(x^2+y^2)^(-1/2)
then f(r,theta) -> f(x,y) = 1 + y*(x^2+y^2)^(-1/2) - (x^2+y^2)^(1/2) =
Good luck solving this for z... better stick with polar coordinates. Most calculators can be set into polar coordinates too.