Trigonometry (10th Edition)

Published by Pearson
ISBN 10: 0321671775
ISBN 13: 978-0-32167-177-6

Chapter 8 - Complex Numbers, Polar Equations, and Parametric Equations - Section 8.3 The Product and Quotient Theorems - 8.3 Exercises - Page 370: 21

Answer

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

Work Step by Step

Now, $-i$ is at $270^\circ$ with unit 1, and $1+i$ is at $45^\circ$ with absolute value $\sqrt{1^2 + 1^2} = \sqrt{2}$ Therefore, $\frac{-i}{1+i}$ = $\frac{1 cis270^\circ}{\sqrt{2}cis45^\circ}$ = $\frac{1}{\sqrt{2}}\cdot cis(270^\circ - 45^\circ)$ (Quotient Theorem) = $\frac{1}{\sqrt{2}}\cdot cis225^\circ$ = $\frac{1}{\sqrt{2}}\cdot (\frac{-1}{\sqrt{2}} + \frac{-1}{\sqrt{2}} i)$ = $- \frac{1}{2} - \frac{1}{2} i$
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