Answer
$r^2 = 4~sin~2\theta$
This graph is a lemniscate.
We can see this graph below:
Work Step by Step
$r^2 = 4~sin~2\theta$
Note that the graph only includes points where $sin~2\theta \geq 0$
That is:
$0 \leq \theta \leq 90^{\circ}$
$180 \leq \theta \leq 270^{\circ}$
When $\theta = 0^{\circ}$, then $r = \sqrt{4~sin~0^{\circ}} = 0$
When $\theta = 15^{\circ}$, then $r = \sqrt{4~sin~30^{\circ}} = 1.41$
When $\theta = 30^{\circ}$, then $r = \sqrt{4~sin~60^{\circ}} = 1.86$
When $\theta = 45^{\circ}$, then $r = \sqrt{4~sin~90^{\circ}} = 2$
When $\theta = 60^{\circ}$, then $r = \sqrt{4~sin~120^{\circ}} = 1.86$
When $\theta = 90^{\circ}$, then $r = \sqrt{4~sin~180^{\circ}} = 0$
When $\theta = 180^{\circ}$, then $r = \sqrt{4~sin~360^{\circ}} = 0$
When $\theta = 225^{\circ}$, then $r = \sqrt{4~sin~450^{\circ}} = 2$
When $\theta = 270^{\circ}$, then $r = \sqrt{4~sin~540^{\circ}} = 0$
This graph is a lemniscate.
We can see this graph below:
