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 8.4 De Moivre's Theorem: Powers and Roots of Complex Numbers - 8.4 Exercises - Page 384: 56


$x = 0.343+1.455~i$ $x = -0.343-1.455~i$

Work Step by Step

$x^2+2-i = 0$ $x^2 = -2+i$ We can find $r$: $r = \sqrt{(-2)^2+(1)^2} = \sqrt{5}$ Note that $~~-2+i~~$ is in the second quadrant: $\theta = tan^{-1}(-\frac{1}{2})+180^{\circ}$ $\theta = -26.565^{\circ}+180^{\circ} = 153.435^{\circ}$ We can find $x$: When $k=0$: $x = (\sqrt{5})^{1/2}(cos~\frac{153.435^{\circ}+360^{\circ}~k}{2}+i~sin~\frac{153.435^{\circ}+360^{\circ}~k}{2})$ $x = (\sqrt{5})^{1/2}(cos~76.72^{\circ}+i~sin~76.72^{\circ})$ $x = 0.343+1.455~i$ When $k=1$: $x = (\sqrt{5})^{1/2}(cos~\frac{153.435^{\circ}+360^{\circ}~k}{2}+i~sin~\frac{153.435^{\circ}+360^{\circ}~k}{2})$ $x = (\sqrt{5})^{1/2}(cos~256.72^{\circ}+i~sin~256.72^{\circ})$ $x = -0.343-1.455~i$
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