Rectangular -> Polar Equation Conversion Help?

Question

I don't understand how to convert r=2sin(3theta) into a rectangular eq. any help would be appreciated.

Answer 1

r = 2sin(3θ)
This is a flower with 3 petals.

r² = 2rsin(3θ) = 2r[3(cos²θ)(sinθ) - sin³θ]
r² = 2rsinθ[2cos²θ - sin²θ]
r² = 2y[2(x/r)² - (y/r)²]
r² = 2y[2(x/r)² - (y/r)²]
r^4 = 4x²y - 2y³
(x² + y²)² = 4x²y - 2y³

Answer 2

Let theta = t

The equation becomes :

r = 2 sin (3t)

Multiply by r on both sides and let 3t be p

r * r = 2 r sin (p)

x^2 + y^2 = 2 y.......using the equations -> x = r cos p &
y = r sin p

x^2 + y^2 - 2y = 0

This is the equation of the circle having radius 1 units and center (0,1)

Answer 3

x = r*cos(θ)
y = r*sin(θ)

r = sqrt(x^2 + y^2)
θ = atan(y/x)
You may simply write the equation as
sqrt(x^2 + y^2) = 2*sin[ 2*atan(y/x) ]

Or you can expand further
sin(3θ) = sin(2θ + θ)
= sin(2θ)*cos(θ) + cos(2θ)*sin(θ)
= 2*sinθ*cos^2 θ + sinθ * (cos^2 θ - sin^2 θ)
= sinθ * (3*cos^2 θ - sin^2 θ)

So r = 2*(y/r) * (3x^2 / r^2 - y^2 / r^2)
r^4 = (x^2 + y^2)^2 = 2y * (3x^2 - y^2)

**EDIT**
This is NOT a circle. Plot it in polar cordinates and you'll end up getting something like a flower with 3 petals.