Trigonometry (11th Edition) Clone

Published by Pearson
ISBN 10: 978-0-13-421743-7
ISBN 13: 978-0-13421-743-7

Chapter 8 - Complex Numbers, Polar Equations, and Parametric Equations - Section 8.5 Polar Equations and Graphs - 8.5 Exercises - Page 395: 53

Answer

$r^2 = 4~cos~2\theta$ This graph is a lemniscate. We can see this graph below:
1555682448

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:
Small 1555682448
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