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: 31

Answer

(a) $x = 3~sin~t$ $y = 2~cos~t$ We can see the graph below. (b) $\frac{x^2}{9}+\frac{y^2}{4} = 1$
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Work Step by Step

(a) $x = 3~sin~t$ $y = 2~cos~t$ When $t = 0$: $x = 3~sin~0 = 0$ $y = 2~cos~0 = 2$ When $t = \frac{\pi}{6}$: $x = 3~sin~\frac{\pi}{6} = \frac{3}{2}$ $y = 2~cos~\frac{\pi}{6} = \sqrt{3}$ When $t = \frac{\pi}{4}$: $x = 3~sin~\frac{\pi}{4} = \frac{3\sqrt{2}}{2}$ $y = 2~cos~\frac{\pi}{4} = \sqrt{2}$ When $t = \frac{\pi}{3}$: $x = 3~sin~\frac{\pi}{3} = \frac{3\sqrt{3}}{2}$ $y = 2~cos~\frac{\pi}{3} = 1$ When $t = \frac{\pi}{2}$: $x = 3~sin~\frac{\pi}{2} = 3$ $y = 2~cos~\frac{\pi}{2} = 0$ When $t = \frac{2\pi}{3}$: $x = 3~sin~\frac{2\pi}{3} = \frac{3\sqrt{3}}{2}$ $y = 2~cos~\frac{2\pi}{3} = -1$ When $t = \pi$: $x = 3~sin~\pi = 0$ $y = 2~cos~\pi = -2$ When $t = \frac{4\pi}{3}$: $x = 3~sin~\frac{4\pi}{3} = -\frac{3\sqrt{3}}{2}$ $y = 2~cos~\frac{4\pi}{3} = -1$ When $t = \frac{3\pi}{2}$: $x = 3~sin~\frac{3\pi}{2} = -3$ $y = 2~cos~\frac{3\pi}{2} = 0$ We can see the graph below. (b) $x = 3~sin~t$ $y = 2~cos~t$ $\frac{x^2}{9}+\frac{y^2}{4} = \frac{(3~sin~t)^2}{9}+\frac{(2~cos~t)^2}{4}$ $\frac{x^2}{9}+\frac{y^2}{4} = \frac{9~sin^2~t}{9}+\frac{4~cos^2~t}{4}$ $\frac{x^2}{9}+\frac{y^2}{4} = sin^2~t+cos^2~t$ $\frac{x^2}{9}+\frac{y^2}{4} = 1$
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