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 8.6 Parametric Equations, Graphs, and Applications - 8.6 Exercises - Page 404: 40

Answer

$x = 3~cos~2t$ $y = 3~sin~3t$ $0 \leq t \leq 6.5$ We can see the graph in the window $[-6,6]$ by $[-4,4]$
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Work Step by Step

$x = 3~cos~2t$ $y = 3~sin~3t$ $0 \leq t \leq 6.5$ When $t = 0$: $x = 3~cos~0 = 3$ $y = 3~sin~0 = 0$ When $t = \frac{\pi}{12}$: $x = 3~cos~\frac{2\pi}{12} = 2.6$ $y = 3~sin~\frac{3\pi}{12} = 2.12$ When $t = \frac{\pi}{6}$: $x = 3~cos~\frac{2\pi}{6} = 1.5$ $y = 3~sin~\frac{3\pi}{6} = 3$ When $t = \frac{\pi}{4}$: $x = 3~cos~\frac{2\pi}{4} = 0$ $y = 3~sin~\frac{3\pi}{4} = 2.12$ When $t = \frac{\pi}{3}$: $x = 3~cos~\frac{2\pi}{3} = -1.5$ $y = 3~sin~\frac{3\pi}{3} = 0$ When $t = \frac{\pi}{2}$: $x = 3~cos~\frac{2\pi}{2} = -3$ $y = 3~sin~\frac{3\pi}{2} = -3$ When $t = \frac{2\pi}{3}$: $x = 3~cos~\frac{4\pi}{3} = -1.5$ $y = 3~sin~\frac{6\pi}{3} = 0$ When $t = \frac{3\pi}{4}$: $x = 3~cos~\frac{6\pi}{4} = 0$ $y = 3~sin~\frac{9\pi}{4} = 2.12$ When $t = \pi$: $x = 3~cos~2\pi = 3$ $y = 3~sin~3\pi = 0$ When $t = \frac{7\pi}{6}$: $x = 3~cos~\frac{14\pi}{6} = 1.5$ $y = 3~sin~\frac{21\pi}{6} = -3$ When $t = \frac{5\pi}{4}$: $x = 3~cos~\frac{10\pi}{4} = 0$ $y = 3~sin~\frac{15\pi}{4} = -2.12$ When $t = \frac{7\pi}{3}$: $x = 3~cos~\frac{14\pi}{3} = -1.5$ $y = 3~sin~\frac{21\pi}{3} = 0$ When $t = \frac{3\pi}{2}$: $x = 3~cos~\frac{6\pi}{2} = -3$ $y = 3~sin~\frac{9\pi}{2} = 3$ When $t = \frac{7\pi}{4}$: $x = 3~cos~\frac{14\pi}{4} = 0$ $y = 3~sin~\frac{21\pi}{4} = -2.12$ When $t = 2\pi$: $x = 3~cos~4\pi = 3$ $y = 3~sin~6\pi = 0$ When $t = 6.5$: $x = 3~cos~13 = 2.72$ $y = 3~sin~19.5 = 1.82$ We can see the graph in the window $[-6,6]$ by $[-4,4]$
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