Book edition 6th

Author(s) Sullivan

Pages 1200 pages

ISBN 9780321795465

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Short Answer

Find zw and $\frac{z}{w}$ . Leave the answers in polar form.
role="math" localid="1646661399572" $z = 2 + 2 i w = \sqrt{3} - i$

zw in polar form is $4 \sqrt{2} (\cos 15 ° + i \sin 15 °)$ .

$\frac{z}{w}$ in polar form is $\sqrt{2} (\cos 75 ° + i \sin 75 °)$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information.

Consider the given question,

$z = 2 + 2 i w = \sqrt{3} - i$

The multiplication of the polar form of complex numbers is given by,

$z_{1} z_{2} = r_{1} r_{2} [\cos (θ_{1} + θ_{2}) + i \sin (θ_{1} + θ_{2})] . . . . . . (i) \frac{z_{1}}{z_{2}} = \frac{r_{1}}{r_{2}} [\cos (θ_{1} - θ_{2}) + i \sin (θ_{1} - θ_{2})] . . . . . . (i i)$

We need to write the given numbers in polar form.

$r = \sqrt{x^{2} + y^{2}}, θ = \tan^{- 1} (\frac{y}{x})$

Step 2. Find the polar form of w.

Determine the polar form of z,

$r_{z} = \sqrt{2^{2} + 2^{2}} r_{z} = 2 \sqrt{2} θ_{z} = \tan^{- 1} (\frac{2}{2}) θ_{z} = 45 °$

Therefore, $z = 2 \sqrt{2} (\cos 45 ° + i \sin 45 °)$ .

Determine the polar form of w,

$r_{w} = \sqrt{{(\sqrt{3})}^{2} + {(- 1)}^{2}} r_{w} = 2 θ_{w} = \tan^{- 1} (\frac{- 1}{\sqrt{3}}) θ_{w} = - 30 °$

Therefore, $w = 2 (\cos (- 30 °) + i \sin (- 30 °))$

Step 3. Find zw and zw.

Substitute the values of z,w in equation (i),

$z \cdot w = 2 \sqrt{2} [\cos (45 ° + i \sin 45 °)] \cdot 2 [\cos (- 30 °) + i \sin (- 30 °)] z \cdot w = (2) 2 \sqrt{2} [\cos (45 ° + (- 30 °)) + i \sin (45 ° + (- 30 °))] z \cdot w = 4 \sqrt{2} (\cos 15 ° + i \sin 15 °)$

Substitute the values of z,w in equation (ii),

$\frac{z}{w} = \frac{2 \sqrt{2} (\cos 45 ° + i \sin 45 °)}{2 (\cos (- 30 °) + i \sin (- 30 °))} \frac{z}{w} = \frac{2 \sqrt{2}}{2} [\cos (45 ° - (- 30 °)) + i \sin (45 ° - (- 30 °))] \frac{z}{w} = \sqrt{2} (\cos 75 ° + i \sin 75 °)$

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Precalculus Enhanced with Graphing Utilities

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Short Answer

Find zw and $\frac{z}{w}$ . Leave the answers in polar form.
role="math" localid="1646661399572" $z = 2 + 2 i w = \sqrt{3} - i$

Step by Step Solution

Step 1. Given information.

Step 2. Find the polar form of w.

Step 3. Find zw and zw.

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Step by Step Solution

Step 1. Given information.

Step 2. Find the polar form of w.

Step 3. Find zw and zw.

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Find zw and $\frac{z}{w}$ . Leave the answers in polar form.
role="math" localid="1646661399572" $z = 2 + 2 i w = \sqrt{3} - i$