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 594: 43

Answer

$-32+32\sqrt3i$

Work Step by Step

RECALL: De Moivre's Theorem: $(\cos{x}+i\sin{x})^n=\cos{(nx)}+i\sin{(nx)}$ Use the theorem above to obtain: $[4(\cos40^\circ+i\sin40^\circ]^3 \\=4^3\left(\cos(3\cdot40^\circ)+i\cdot \sin(3\cdot40^\circ)\right) \\=64\cdot(\cos(120^\circ)+i(\sin(120^\circ)) \\=64\left(-\frac{1}{2}+i\frac{\sqrt3}{2}\right) \\=-32+32\sqrt3i$
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