Algebra and Trigonometry 10th Edition

Published by Cengage Learning
ISBN 10: 9781337271172
ISBN 13: 978-1-33727-117-2

Chapter 8 - 8.6 - Trigonometric Form of a Complex Number - 8.6 Exercises - Page 613: 54


$[3(cos~60°+i~sin~60°)]^4=-\frac{81}{2}-\frac{81\sqrt 3}{2}i$

Work Step by Step

DeMoivre's Theorem: If $z=r(cos θ+i~sin θ)$, then $z^n=r^n(cos~nθ+i~sin~nθ)$ $z=3(cos~60°+i~sin~60°)$ $z^4=3^4[cos~(4 ·60°)+i~sin~(4 ·60°)]$ $z^4=81(cos~240°+i~sin~240°)$ $z^4=81[-\frac{1}{2}+i(-\frac{\sqrt 3}{2})]$ $z^4=-\frac{81}{2}-\frac{81\sqrt 3}{2}i$
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