Trigonometry (11th Edition) Clone

Published by Pearson
ISBN 10: 978-0-13-421743-7
ISBN 13: 978-0-13421-743-7

Chapter 8 - Complex Numbers, Polar Equations, and Parametric Equations - Section 8.3 The Product and Quotient Theorems - 8.3 Exercises - Page 376: 26

Answer

$\frac{1}{2 - 2i}$ = $\frac{1}{4} + \frac{1}{4} i$

Work Step by Step

Now, 1 is at $0^\circ$ with unit 1, and, $2 - 2i$ is at $315^\circ$ with absolute value $\sqrt{2^2 + (-2)^2} = \sqrt{8}$ Therefore, $\frac{1}{2 - 2i}$ = $\frac{1cis0^\circ}{\sqrt{8}cis315^\circ}$ = $\frac{1}{\sqrt{8}} cis(0^\circ - 315^\circ)$ (Quotient Theorem) = $\frac{1}{\sqrt{8}} cis(-315^\circ)$ = $\frac{1}{\sqrt{8}} (\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} i)$ = $\frac{1}{4} + \frac{1}{4} i$
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