# Chapter 8 - Complex Numbers, Polar Equations, and Parametric Equations - Section 8.5 Polar Equations and Graphs - 8.5 Exercises - Page 395: 54

$r^2 = 4~sin~2\theta$ This graph is a lemniscate. We can see this graph below: $r^2 = 4~sin~2\theta$ 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: 