University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 10 - Section 10.6 - Conics in Polar Coordinates - Exercises - Page 591: 45

Answer

$x+y=2$ .

Work Step by Step

Apply identity for $\cos(\alpha-\beta)$ $\displaystyle \cos(\theta-\frac{\pi}{4})=\cos\theta\cos\frac{\pi}{4}+\sin\theta\sin\frac{\pi}{4}$ $\displaystyle \cos(\theta-\frac{\pi}{4})=\frac{1}{\sqrt{2}}\cdot\cos\theta+\frac{1}{\sqrt{2}}\cdot\sin\theta\qquad/\times r$ $r\displaystyle \cos(\theta-\frac{\pi}{4})=\frac{1}{\sqrt{2}}\cdot(r\cos\theta)+\frac{1}{\sqrt{2}}\cdot(r\sin\theta)$ Apply the conversion formula$\quad (x,y)=(r\cos\theta,r\sin\theta)$ $r\displaystyle \cos(\theta-\frac{\pi}{4})=\frac{1}{\sqrt{2}}\cdot x+\frac{1}{\sqrt{2}}\cdot y$ In Cartesian coordinates, the line equation is $\displaystyle \frac{1}{\sqrt{2}}\cdot x+\frac{1}{\sqrt{2}}\cdot y=\sqrt{2}\qquad/\times\sqrt{2}$ $x+y=2$
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