Express this polar equation in cartesian form

Question

Having trouble converting this polar equation into Cartesian form:

$r = 2 + \sin(\theta)$

This is how far I get:

$(r = 2 + \sin(\theta))\cdot r$

$r^2 = 2r + r\sin(\theta)$

$x^2 + y^2 = 2r + y$, since $r^2 = x^2 + y^2$ and $y = r\sin(\theta)$

Where do I go from here or where did I go wrong? Thanks

Answer 1

If $r=2+\sin \theta$, as $r^2=x^2+y^2$ and $y=r\sin \theta$, then multiplying the equation by $r$ we obtain $$ r^2=2r+r\sin \theta ,$$ and then $$x^2+y^2=2\sqrt{x^2+y^2}+y, $$ which is a cardioid.

Hope this helps.