Question

find the eccentricity to identify the conic, then sketch the conic and its directrix

r=1/1-cos theta

r=2/3+2 sin theta

r=4/1-2 cos theta

Answer 1

The general form is

r = k/(1 ± e cosθ)
or
r = k/(1 ± e sinθ)

so, you need to massage what you have into that form

2/(3+2sinθ) = (2/3)/(1 + 2/3 sinθ)
so, e = 2/3

do the others in like wise.