Why can't I graph r= 1 + 2cosθ by converting it into rectangular coordinates

Question

I'm trying to graph the equation r = 1 + $2\cos \theta$ using its rectangular form.

I converted this equation to rectangular coordinates like this:

$$\sqrt{x^2+y^2} = 1 + 2 \cos\Biggl(\arccos\left(\frac{x}{\sqrt{ (x^2+y^2)}}\right)\Biggl)$$

When I graph them I get:

This graph for the cartesian form

This graph for the polar form

The petal shape is not present in the cartesian form. What am I doing wrong here?

Answer 1

The inverse cosine function is tricky.

Try parametric equations to graph the curve. $$x=(1+2\cos(\theta))\cos(\theta), y=(1+2\cos(\theta)\sin(\theta))$$

You should see the whole picture.