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 397: 84b

Answer

$r^2 = 22,500~sin~2\theta$ The signal can be received in locations that are northeast or southwest of the origin up to a distance of 150 miles away. The signal can be received in regions which are within $45^{\circ}$ of the northeast-southwest line. The signal can not be received in any locations northwest or southeast of the origin, or regions located within $45^{\circ}$ of the northwest-southeast line.
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Work Step by Step

$r^2 = 22,500~sin~2\theta$ Note that the graph only includes points where $sin~2\theta \geq 0$ That is: $0 \leq \theta \leq 90^{\circ}$ $180 \leq \theta \leq 270^{\circ}$ When $\theta = 0^{\circ}$, then $r = \sqrt{22,500~sin~0^{\circ}} = 0$ When $\theta = 15^{\circ}$, then $r = \sqrt{22,500~sin~30^{\circ}} = 106$ When $\theta = 30^{\circ}$, then $r = \sqrt{22,500~sin~60^{\circ}} = 139.6$ When $\theta = 45^{\circ}$, then $r = \sqrt{22,500~sin~90^{\circ}} = 150$ When $\theta = 60^{\circ}$, then $r = \sqrt{22,500~sin~120^{\circ}} = 139.6$ When $\theta = 90^{\circ}$, then $r = \sqrt{22,500~sin~180^{\circ}} = 0$ When $\theta = 180^{\circ}$, then $r = \sqrt{22,500~sin~360^{\circ}} = 0$ When $\theta = 225^{\circ}$, then $r = \sqrt{22,500~sin~450^{\circ}} = 150$ When $\theta = 270^{\circ}$, then $r = \sqrt{22,500~sin~540^{\circ}} = 0$ We can see this graph below. Since locations where the signal can be received correspond to the interior of the curve, the signal can be received in locations that are northeast or southwest of the origin up to a distance of 150 miles away. Note that the signal can be received in regions which are within $45^{\circ}$ of the northeast-southwest line, that is, in the first and third quadrants. The signal can not be received in regions outside of the curve, so the signal can not be received in any locations northwest or southeast of the origin, or regions located within $45^{\circ}$ of the northwest-southeast line, that is, in the second and fourth quadrants.
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