convert the polar equation to its rectangular equivalent

Question

r = 4sin (theta)

Answer 1

sin (theta) = y/r
so
r = 4 y / r
[ sqrt (x^2+y^2) ]^2 = 4 y
x^2 + y^2 = 4 y
y^2 - 4 y = -x^2
y^2 -4 y + 4 = -x^2 +4
(y-2)^2 =-(x^2-4)
y-2 = sqrt (4-x^2)
y = 2 - sqrt (4-x^2)

Answer 2

(y-2)^2 =-(x^2-4)
y-2 = or -sqrt (4-x^2)
y = 2 - sqrt (4-x^2)
or y = 2 + sqrt (4-x^2)

Answer 3

r = 4sinθ
r^2 = 4r sinθ
x^2+y^2 = 4y
x^2 + y^2 - 4y = 0
x^2 + y^2 - 4y + 4 = 4
x^2 + (y-2)^2 = 4