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: 53


$[5(cos~20°+i~sin~20°)]^3=\frac{125}{2}+\frac{125\sqrt 3}{2}i$

Work Step by Step

DeMoivre's Theorem: If $z=r(cos θ+i~sin θ)$, then $z^n=r^n(cos~nθ+i~sin~nθ)$ $z=5(cos~20°+i~sin~20°)$ $z^3=5^3[cos~(3 ·20°)+i~sin~(3 ·20°)]$ $z^3=125(cos~60°+i~sin~60°)$ $z^3=125(\frac{1}{2}+i\frac{\sqrt 3}{2})$ $z^3=\frac{125}{2}+\frac{125\sqrt 3}{2}i$
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