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: 36


$zw=(\cos({220^\circ)}+i\sin{(220^\circ})$ $\frac{z}{w}=(\cos({20^\circ)}+i\sin{20^\circ})$.

Work Step by Step

We know that if $z=a(\cos{\alpha}+i\sin{\alpha})$ and $w=b(\cos{\beta}+i\sin{\beta})$, then $zw=ab(\cos({\alpha+\beta)}+i\sin{(\alpha+\beta})$ and $\frac{z}{w}=\frac{a}{b}(\cos({\alpha-\beta)}+i\sin{(\alpha-\beta})$. Hence here: $zw=(1)(1)(\cos({120^\circ+100^\circ)}+i\sin{(120^\circ+100^\circ})\\zw=(\cos({220^\circ)}+i\sin{(220^\circ})$ and $\frac{z}{w}=\frac{1}{1}(\cos({120^\circ-100^\circ)}+i\sin{120^\circ-100^\circ})\\\frac{z}{w}=(\cos({20^\circ)}+i\sin{20^\circ})$.
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