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 8.1 Complex Numbers - 8.1 Exercises - Page 358: 83

Answer

$$\frac{2-i}{2+i}=\frac{3}{5}-\frac{4}{5}i$$

Work Step by Step

$$A=\frac{2-i}{2+i}$$ For fraction operation, we need to multiply the complex conjugate of the denominator in both the numerator and the denominator. In other words. here we need to multiply both the numerator and denominator by $(2-i)$ $$A=\frac{2-i}{2+i}\frac{2-i}{2-i}$$ $$A=\frac{(2-i)^2}{4-i^2}$$ (for $(a-b)(a+b)=a^2-b^2$) $$A=\frac{4+i^2-4i}{4-(-1)}$$ (for $(a-b)^2=a^2+b^2-2ab$)$$A=\frac{4-1-4i}{4+1}$$ $$A=\frac{3-4i}{5}$$ $$A=\frac{3}{5}-\frac{4}{5}i$$
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