Answer
$r^2 = 4~sin~2\theta$
This graph is a lemniscate.
We can see this graph below:
![](https://gradesaver.s3.amazonaws.com/uploads/solution/1500f660-662e-492f-9733-73eac57f6fa9/result_image/1555683088.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAJVAXHCSURVZEX5QQ%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T011655Z&X-Amz-Expires=900&X-Amz-SignedHeaders=host&X-Amz-Signature=f4d405120bc14cb8308096219dbf578b38603782bd03a71e2b6027e58a2ae905)
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:
![](https://gradesaver.s3.amazonaws.com/uploads/solution/1500f660-662e-492f-9733-73eac57f6fa9/steps_image/small_1555683088.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAJVAXHCSURVZEX5QQ%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T011655Z&X-Amz-Expires=900&X-Amz-SignedHeaders=host&X-Amz-Signature=b0645046a8399197a47efd763898b4a3f0292d997bdf8474503174c03fa04d1c)