Answer
$r^2 = 4~cos~2\theta$
This graph is a lemniscate.
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Work Step by Step
$r^2 = 4~cos~2\theta$
Note that the graph only includes points where $cos~2\theta \geq 0$
That is:
$0 \leq \theta \leq 45^{\circ}$
$135 \leq \theta \leq 225^{\circ}$
$315 \leq \theta \leq 360^{\circ}$
When $\theta = 0^{\circ}$, then $r = \sqrt{4~cos~0^{\circ}} = 2$
When $\theta = 15^{\circ}$, then $r = \sqrt{4~cos~30^{\circ}} = 1.86$
When $\theta = 30^{\circ}$, then $r = \sqrt{4~cos~60^{\circ}} = 1.41$
When $\theta = 45^{\circ}$, then $r = \sqrt{4~cos~90^{\circ}} = 0$
When $\theta = 135^{\circ}$, then $r = \sqrt{4~cos~270^{\circ}} = 0$
When $\theta = 150^{\circ}$, then $r = \sqrt{4~cos~300^{\circ}} = 1.41$
When $\theta = 180^{\circ}$, then $r = \sqrt{4~cos~360^{\circ}} = 2$
When $\theta = 225^{\circ}$, then $r = \sqrt{4~cos~450^{\circ}} = 0$
When $\theta = 315^{\circ}$, then $r = \sqrt{4~cos~630^{\circ}} = 0$
When $\theta = 330^{\circ}$, then $r = \sqrt{4~cos~660^{\circ}} = 1.41$
When $\theta = 345^{\circ}$, then $r = \sqrt{4~cos~690^{\circ}} = 1.86$
When $\theta = 360^{\circ}$, then $r = \sqrt{4~cos~720^{\circ}} = 2$
This graph is a lemniscate.
We can see this graph below:
