Answer
$r^2 = 4~cos~2\theta$
This graph is a lemniscate.
We can see this graph below:
![](https://gradesaver.s3.amazonaws.com/uploads/solution/385f07ee-a027-461f-803a-02d3ca75fdbc/result_image/1555682448.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAJVAXHCSURVZEX5QQ%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T013135Z&X-Amz-Expires=900&X-Amz-SignedHeaders=host&X-Amz-Signature=05639e0742f607e0aae69612a5fff322c07036628a4d5a109eeee3afa53359cf)
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:
![](https://gradesaver.s3.amazonaws.com/uploads/solution/385f07ee-a027-461f-803a-02d3ca75fdbc/steps_image/small_1555682448.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAJVAXHCSURVZEX5QQ%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T013135Z&X-Amz-Expires=900&X-Amz-SignedHeaders=host&X-Amz-Signature=00aa4d33ee64507c5147dad941fd5e597ba9bbd072ab637279e73394bde9e88c)