Trigonometry 7th Edition

Published by Cengage Learning
ISBN 10: 1111826854
ISBN 13: 978-1-11182-685-7

Chapter 8 - Section 8.6 - Equations in Polar Coordinates and Their Graphs - 8.6 Problem Set - Page 468: 59

Answer

$r^2 = \sin 2\theta$

Work Step by Step

we know $x = r \cos \theta$ and $y = r \sin \theta$ putting the values in equation we get ${(x^2 +y^2)}^2 = 2xy$ $=> {({(r \cos \theta)}^2 + {(r \sin \theta)}^2)}^2 =2r^2 \cos \theta \sin\theta$ $=> r^4{({ \cos^2 \theta} + {\sin^2 \theta})}^2=2r^2 \cos \theta \sin\theta$ We know $({ \cos^2 \theta} + {\sin^2 \theta}) = 1$ and $2\cos \theta \sin \theta = \sin 2\theta$ $=> r^4 = r^2 \sin 2\theta$ $=> r^2 = \sin 2\theta$ The graph of the above equation is two-leaved rose as shown
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