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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. 35 (page 592)

In Problems $33 - 40$ , find $z w$ and $\frac{z}{w}$ .Leave your answers in polar form.
$z = 3 (\cos 130^{鈭�} + i \sin 130^{鈭�}) w = 4 (\cos 270^{鈭�} + i \sin 270^{鈭�})$

Short Answer

Expert verified

The polar form of $z w$ is, role="math" localid="1646761395771" $12 [\cos 40^{鈭�} + i \sin 40^{鈭�}]$ .

The polar form of $\frac{z}{w}$ is, $\frac{3}{4} [\cos 220^{鈭�} + i \sin 220^{鈭�}]$ .

Step by step solution

01

Step 1 Given that,

$z = 3 (\cos 130^{鈭�} + i \sin 130^{鈭�}) w = 4 (\cos 270^{鈭�} + i \sin 270^{鈭�})$

First find $z w$ .

$z . w = 3 (\cos 130^{鈭�} + i \sin 130^{鈭�}) . 4 (\cos 270^{鈭�} + i \sin 270^{鈭�}) = 12 [\cos (130^{鈭�} + 270^{鈭�}) + i \sin (130^{鈭�} + 270^{鈭�})] = 12 [\cos 400^{鈭�} + i \sin 400^{鈭�}]$

we know that , $\cos (360^{鈭�} + 胃) = \cos 胃,$ so $\cos (360^{鈭�} + 40^{鈭�}) = \cos 40^{鈭�}$ .

$z . w = 12 [\cos 40^{鈭�} + i \sin 40^{鈭�}]$ .

02

Step 2 Next find zw.

$\frac{z}{w} = \frac{3 (\cos 130^{鈭�} + i \sin 130^{鈭�})}{4 (\cos 270^{鈭�} + i \sin 270^{鈭�})} = \frac{3}{4} [\cos (130^{鈭�} - 270^{鈭�}) + i \sin (130^{鈭�} - 270^{鈭�})] = \frac{3}{4} [\cos (- 140^{鈭�}) + i \sin (- 140^{鈭�})]$

we know that,

\cos (- 140^{鈭�}) = \cos (360^{鈭�} - 220^{鈭�}) = \cos 220^{鈭�}

Thus, $\frac{z}{w} = \frac{3}{4} [\cos 220^{鈭�} + i \sin 220^{鈭�}]$ .

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Most popular questions from this chapter

In Problems 13鈥�28, transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. Verify your graph using a graphing utility.

$r \cos 胃 = 4$

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

$r^{2} = \sin (2 胃)$

Find a vector $v$ whose magnitude is $4$ and whose component

in the $i$ direction is twice the component in the $j$ direction.

find the unit vector in the same direction as $v$

$v = - 3 j$

$v = - 3 j$

Find each quantity if $v = 3 i - 5 j$ and $w = - 2 i + 3 j$ .

$3 v - 2 w$

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