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: 42

Answer

$x = 4~sin~4t$ $y = 3~sin~5t$ $0 \leq t \leq 6.5$ We can see the graph in the window $[-6,6]$ by $[-4,4]$
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Work Step by Step

$x = 4~sin~4t$ $y = 3~sin~5t$ $0 \leq t \leq 6.5$ When $t = 0$: $x = 4~sin~0 = 0$ $y = 3~sin~0 = 0$ When $t = \frac{\pi}{12}$: $x = 4~sin~\frac{4\pi}{12} = 3.46$ $y = 3~sin~\frac{5\pi}{12} = 2.90$ When $t = \frac{\pi}{6}$: $x = 4~sin~\frac{4\pi}{6} = 3.46$ $y = 3~sin~\frac{5\pi}{6} = 1.5$ When $t = \frac{\pi}{5}$: $x = 4~sin~\frac{4\pi}{5} = 2.35$ $y = 3~sin~\frac{5\pi}{5} = 0$ When $t = \frac{\pi}{4}$: $x = 4~sin~\frac{4\pi}{4} = 0$ $y = 3~sin~\frac{5\pi}{4} = -2.12$ When $t = \frac{\pi}{3}$: $x = 4~sin~\frac{4\pi}{3} = - 3.46$ $y = 3~sin~\frac{5\pi}{3} = -2.60$ When $t = \frac{2\pi}{5}$: $x = 4~sin~\frac{8\pi}{5} = -3.8$ $y = 3~sin~\frac{10\pi}{5} = 0$ When $t = \frac{\pi}{2}$: $x = 4~sin~\frac{4\pi}{2} = 0$ $y = 3~sin~\frac{5\pi}{2} = 3$ When $t = \frac{2\pi}{3}$: $x = 4~sin~\frac{8\pi}{3} = 3.46$ $y = 3~sin~\frac{10\pi}{3} = -2.6$ When $t = \frac{3\pi}{4}$: $x = 4~sin~\frac{12\pi}{4} = 0$ $y = 3~sin~\frac{15\pi}{4} = -2.12$ When $t = \frac{4\pi}{5}$: $x = 4~sin~\frac{16\pi}{5} = -2.35$ $y = 3~sin~\frac{20\pi}{5} = 0$ When $t = \frac{5\pi}{6}$: $x = 4~sin~\frac{20\pi}{6} = - 3.46$ $y = 3~sin~\frac{25\pi}{6} = 1.5$ When $t = \pi$: $x = 4~sin~4\pi = 0$ $y = 3~sin~5\pi = 0$ When $t = \frac{7\pi}{6}$: $x = 4~sin~\frac{28\pi}{6} = 3.46$ $y = 3~sin~\frac{35\pi}{6} = -1.5$ When $t = \frac{5\pi}{4}$: $x = 4~sin~\frac{20\pi}{4} = 0$ $y = 3~sin~\frac{25\pi}{4} = 2.12$ When $t = \frac{7\pi}{3}$: $x = 4~sin~\frac{28\pi}{3} = - 3.46$ $y = 3~sin~\frac{35\pi}{3} = 2.60$ When $t = \frac{3\pi}{2}$: $x = 4~sin~\frac{12\pi}{2} = 0$ $y = 3~sin~\frac{15\pi}{2} = -3$ When $t = \frac{7\pi}{4}$: $x = 4~sin~\frac{28\pi}{4} = 0$ $y = 3~sin~\frac{35\pi}{4} = 2.12$ When $t = 2\pi$: $x = 4~sin~8\pi = 0$ $y = 3~sin~10\pi = 0$ When $t = 6.5$: $x = 4~sin~26 = 3.05$ $y = 3~sin~32.5 = 2.65$ We can see the graph in the window $[-6,6]$ by $[-4,4]$
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