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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. 44 (page 592)

Write the expression in the standard form $a + i b$ .
${[3 (\cos 80 掳 + i \sin 80 掳)]}^{3}$

Short Answer

Expert verified

The expression in the standard form is $- 2 \sqrt{2} - 2 i \sqrt{2}$ .

Step by step solution

01

Step 1. Given information.

Consider the given question,

${[3 (\cos 80 掳 + i \sin 80 掳)]}^{3}$

We know that for a complex number raise to n when it is in the polar form $z^{n} = r^{n} (\cos 胃 + i \sin (n 胃)) . . . . . . (i)$

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

$z^{4} = {[\sqrt{2} (\cos \frac{5 蟺}{16} + i \sin \frac{5 蟺}{16})]}^{4}$

02

Step 2. Use equation (i) and solve the equation.

Using the equation (i),

$z^{4} = \cos (4 路 \frac{5 蟺}{16}) + i \sin (4 路 \frac{5 蟺}{16}) z^{4} = 4 (\cos \frac{5 蟺}{4} + i \sin \frac{5 蟺}{4}) z^{4} = 4 [- \frac{\sqrt{2}}{2} + i (- \frac{\sqrt{2}}{2})] z^{4} = - 2 \sqrt{2} - 2 i \sqrt{2}$

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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.

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

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