Converting between Cartesian and polar equations and their graphs

Question

I am asked to convert $(x+2)^2 + y^2 = 4$ into a polar equation, and then confirm on my calculator. I converted it to $r=4cos(\theta)$, but the graphs don’t look the same at all. If anyone could explain how I could verify whenever I get an answer, and why the graphs don’t look the same, that would be great.

Answer 1

There's a sign error.
$(x+2)^2+y^2=4 \implies (x^2+y^2)+4x = 0 \implies r=0,r=\color{blue}{-4\cos\theta}$

Graph in Cartesian coordinates

Graph in Polar coordinates

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For your question in the comment,

EDIT: Just found that $\tan\theta$ and $\sec\theta$ don't work.

let $x=r\cos\theta$ and $y=r\sin\theta$
$ \implies r^2(\cos^2\theta-\sin^2\theta)=r^2\cos2\theta = 4 \implies r = \pm 2\sqrt{\sec2\theta}$