Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 10 - Section 10.3 - Polar Coordinates - 10.3 Exercises - Page 666: 20


The Cartesian form of the expression is $\frac{1}{2x}=y$, which is a rotated hyperbola.

Work Step by Step

We are given that $r^{2}sin2\theta=1$. Using the double-angle identity for sine, we can rewrite the expression as $r^{2}(2sin\theta cos\theta)=1$. We can rewrite this expression as $2(rsin\theta)(rcos\theta)=1$. Using the polar-to-cartesian substitutions of $x=rcos\theta$ and $y=rsin\theta$, we get $2xy=1$. Solving for y, we get $\frac{1}{2x}=y$ and we can see that the graph is that of a scaled reciprocal function, which is a rotated hyperbola.
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