What is the rectangular equation of r=sin3theta?

Question

and of r=3sintheta?

Answer 1

Use the facts that sin(θ) = y/r and r = √(x^2 + y^2). First, for r = 3sin(θ):

r = 3sin(θ)
r = 3(y/r)
r^2 = 3(y)
(x^2 + y^2) = 3y

Now get it into something that looks familar:

y^2 - 3y = -x^2
y^2 - 3y + 9/4 = -x^2 + 9/4
(y - 3/2)^2 = -x^2 + 9/4
(y - 3/2)^2 + x^2 = 9/4. This is a circle.

In the subject line, I'll assume you mean r = sin(3θ) and not r = sin^3 (θ). You'll have to use a trig identity for multiple-angles:
r = sin(3θ)
r = 3cos^2 (x) sin(x) - sin^3(x)
r = 3(x/r)^2 (y/r) - (y/r)^3
r^4 = 3(x)^2 (y) - (y)^3
(x^2 + y^2)^2 = y [ 3(x)^2 - (y)^2 ]