When sketching polar curves, how do you find a restricted domain for θ such that you draw the complete curve once without overlapping?

Answer 1

It depends very much on the nature of the curve and in particular its defining equation. In general, if the curve involves a periodic function of θ, you'll want to just sketch the curve over its period. For example, the curve

r = cos(2θ)

is periodic in θ of period π. Thus the range of θ over which you would sketch the curve would be from θ = 0 to θ = π (any other interval of width π would work just as well).

If your curve is not periodic, for example the logarithmic spiral defined by

r = θ

then θ can take on any value from 0 to ∞.

These are general guidelines, not cut-and-dried answers. It really does depend...