Conversion of the polar equation $ r=\sin(4\theta) + 2$ into Cartesian.

Question

Can some one give me a hand converting $r= \sin(4\theta) +2$ into an x,y equation?

Answer 1

Notice

$$ r = \sin ( 4 \theta ) + 2 = 2 \sin ( 2 \theta ) \cos ( 2 \theta ) + 2 = 4 \sin \theta \cos \theta ( \cos^2 \theta - \sin^2 \theta) + 2 $$

using $x = r \sin \theta$, $y = r \cos \theta $ and $x^2 + y^2 = r^2 $, we have

$$ \boxed{\sqrt{ x^2 + y^2} = 4 \frac{xy(x^2-y^2)}{(x^2+y^2)^2}+ 2}$$