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 382: 13


$-16\sqrt 3+16i$

Work Step by Step

Since, $(\sqrt 3+i)=2(\cos 30^{\circ}+i \sin 30^{\circ})$ and $(\sqrt 3+i)^{5}=2(\cos 30^{\circ}+i \sin 30^{\circ})^{5}$ De Moivre’s Theorem states that when $ r (\cos\theta+i \sin\theta)$ is a complex number, and if $n$ is any real number, then the following relationship holds. $[ r (\cos\theta+i \sin\theta)]^{n}=[ r^{n} (\cos n\theta+i \sin n\theta)]$ In compact form, this is written $[ r cis\theta]^{n}=[ r^{n} (cis \theta)]$ $2(\cos 30^{\circ}+i \sin 30^{\circ})^{5}=2^{5}(\cos 5\times30^{\circ}+i \sin 5\times30^{\circ})^{5}$ $=32(\cos 150^{\circ}+i \sin 150^{\circ})$ $=[3 2(\frac{-\sqrt 3}{2}+i.\frac{ 1}{2})]$ $=-16\sqrt 3+16i$
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