Precalculus (10th Edition)

Published by Pearson
ISBN 10: 0-32197-907-9
ISBN 13: 978-0-32197-907-0

Chapter 9 - Polar Coordinates; Vectors - 9.3 The Complex Plane; De Moivre's Theorem - 9.3 Assess Your Understanding - Page 595: 50



Work Step by Step

Using De Moivre's Theorem ($(\cos{x}+i\sin{x})^n=\cos{nx}+i\sin{nx}.$) and a calculator: $[\sqrt3(\cos{(\frac{5\pi}{18})}+i(\sin{(\frac{5\pi}{18})})]^6=\sqrt3^6(\cos{(6\cdot\frac{5\pi}{18})}+i(\sin{(6\cdot\frac{5\pi}{18})}))=27\cdot(\cos{(\frac{5\pi}{3})}+i(\sin{(\frac{5\pi}{3}})))=27\cdot(\frac{1}{2}+\frac{-\sqrt3}{2}i)=\frac{27}{2}\frac{-27\sqrt3}{2}i.$
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