Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 6 - Section 6.3 - Polar Coordinates - Exercise Set - Page 742: 23

Answer

(a)$$(10, \frac{11\pi }{4})$$ (b)$$(-10, \frac{7\pi }{4})$$ (c)$$(10, -\frac{5\pi }{4})$$
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Work Step by Step

To plot the point $(r, \theta )=(10, \frac{3\pi }{4})$, begin with the $\frac{3\pi }{4}$ angle. Because $\frac{3\pi }{4}$ is a positive angle, draw $\theta = \frac{3\pi }{4}$ counterclockwise from the polar axis. Now consider $r=10$. Because $r \gt 0$, plot the point by going out ten units on the terminal side of $\theta$. Please note that if $n$ is any integer, the point $(r, \theta )$ can be represented as$$(r, \theta ) = (r, \theta +2n\pi ) \\ \text{or} \\ (r, \theta )= (-r, \theta +\pi + 2n\pi ).$$So we have (a)$$(10, \frac{3\pi }{4})= (10, \frac{3\pi }{4}+2(1)\pi)=(10, \frac{11\pi }{4}),$$(b)$$(10, \frac{3\pi }{4})= (-10, \frac{3\pi }{4}+\pi +2(0)\pi )=(-10, \frac{7\pi }{4}),$$(c)$$(10, \frac{3\pi }{4})= (10, \frac{3\pi }{4}+2(-1)\pi)=(10, -\frac{5\pi }{4}).$$
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