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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. 36 (page 592)

In $33 - 40$ Problems , find $z w$ and $\frac{z}{w}$ .Leave your answers in polar form.
$z = 2 (\cos 80^{鈭�} + i \sin 80^{鈭�}) w = 6 (\cos 200^{鈭�} + i \sin 200^{鈭�})$

Short Answer

Expert verified

The polar form of $z w$ is, localid="1646762477334" $z w = 12 [\cos 280^{鈭�} + i \sin 280^{鈭�}]$ .

The polar form of $\frac{z}{w}$ is, $\frac{z}{w} = \frac{1}{3} [\cos 120^{鈭�} - i \sin 120^{鈭�}]$

Step by step solution

01

Step 1 Given that,

$z = 2 (\cos 80^{鈭�} + i \sin 80^{鈭�}) w = 6 (\cos 200^{鈭�} + i \sin 200^{鈭�})$

First find $z w$ .

$z w = 2 (\cos 80^{鈭�} + i \sin 80^{鈭�}) . 6 (\cos 200^{鈭�} + i \sin 200^{鈭�}) = 12 [\cos (80^{鈭�} + 200^{鈭�}) + i \sin (80^{鈭�} + 200^{鈭�})] = 12 [\cos 280^{鈭�} + i \sin 280^{鈭�}]$

02

Step 2 Next find zw.

$\frac{z}{w} = \frac{2 (\cos 80^{鈭�} + i \sin 80^{鈭�})}{6 (\cos 200^{鈭�} + i \sin 200^{鈭�})} = \frac{1}{3} [\cos (80^{鈭�} - 200^{鈭�}) + i \sin (80^{鈭�} - 200^{鈭�})] = \frac{1}{3} [\cos (- 120^{鈭�}) + i \sin (- 120^{鈭�})]$

we know that, $\cos (- 胃) = \cos 胃 \sin (- 胃) = - \sin 胃$

Thus, $\frac{z}{w} = \frac{1}{3} [\cos 120^{鈭�} - i \sin 120^{鈭�}]$

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Most popular questions from this chapter

True or False A cardiod passes through the pole.

Use the figure at the right. Determine whether the given statement is true or false.

$A + B = F$

Write the expression in the standard form $a + i b$ .

${[5 (\cos \frac{3 蟺}{16} + i \sin \frac{3 蟺}{16})]}^{4}$

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

$r = 1 + \sin 胃$

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

$r^{2} = \sin (2 胃)$

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