Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (3rd Edition)

Published by Pearson
ISBN 10: 0-32193-104-1
ISBN 13: 978-0-32193-104-7

Chapter 8 - Polar Coordinates; Vectors - Section 8.3 The Complex Plane; De Moivre's Theorem - 8.3 Assess Your Understanding - Page 615: 33

Answer

$zw=8\left(\cos{60^\circ}+i\sin{60^\circ}\right)$ and $\dfrac{z}{w}=\dfrac{1}{2}\left(\cos{20^\circ}+i\sin{20^\circ}\right)$

Work Step by Step

Let us consider two complex numbers $z=a(\cos{\alpha}+i\sin{\alpha})$ and $w=b(\cos{\beta}+i\sin{\beta})$ Their product and quotient can be expressed as: $zw=ab(\cos({\alpha+\beta)}+i\sin{(\alpha+\beta})$ $\dfrac{z}{w}=\dfrac{a}{b}[\cos({\alpha-\beta)}+i\sin{(\alpha-\beta}]$ Therefore, $zw=(2 \cdot4)(\cos({40^\circ+20^\circ)}+i\sin{(40^\circ+20^\circ})]\\ zw=8\left(\cos{60^\circ}+i\sin{60^\circ}\right)$ and $\dfrac{z}{w}=\dfrac{2}{4}\left[\cos{(40^\circ-20^\circ)}+i\sin{(40^\circ-20^\circ})\right]\\$ $\dfrac{z}{w}=\dfrac{1}{2}\left(\cos{20^\circ}+i\sin{20^\circ}\right)$
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