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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 9: Q. 7 (page 591)

Let $Z_{1} = r_{1} (\cos 胃_{1} + i \sin 胃_{1})$ and $Z_{2} = r_{2} (\cos 胃_{2} + i \sin 胃_{2})$
be two complex numbers. Then
$Z_{1} Z_{2} =__[\cos (_) + i \sin (____)]$

Short Answer

Expert verified

$Z_{1} Z_{2} = r_{1} r_{2} [\cos (胃_{1} + 胃_{2}) + i \sin (胃_{1} + 胃_{2})]$

Step by step solution

01

Step 1. Given

Given complex number in polar form

$z = r \cos 胃 + i \sin 胃$

02

Calculation

$Z_{1} Z_{2} = r_{1} (\cos 胃_{1} + i \sin 胃_{1}) 脳 r_{2} (\cos 胃_{2} + i \sin 胃_{2}) = r_{1} r_{2} (\cos 胃_{1} \cos 胃_{2} + i \cos 胃_{1} \sin 胃_{2} + i \sin 胃_{1} \cos 胃_{2} + i^{2} \sin 胃_{1} \sin 胃_{2}) = r_{1} r_{2} [(\cos (胃_{1} + 胃_{2}) + i \sin (胃_{1} + 胃_{2})]$

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Most popular questions from this chapter

Rose curves are characterized by equations of the form $r = a \cos (n 胃) o r r = a \sin (n 胃), a 鈮� 0 .$ If $n 鈮� 0$ is even, the rose has _____ petals; if $n 鈮� 卤 1$ is odd, the rose has ______ petals.

In Problems55鈥�66, the rectangular coordinates of a point are given. Find polar coordinates for each point.

$(0, 2)$

Identify and graph each polar equation. Verify your graph using a graphing utility

$r = 3 - 3 \sin 胃$

Find zw and $\frac{z}{w}$ . Leave the answers in polar form.

$z = 1 - i w = 1 - \sqrt{3} i$

In Problems 39鈥�54, the polar coordinates of a point are given. Find the rectangular coordinates of each point.

$(8.1, 5.2)$

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