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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. 48 (page 592)

Write the expression in the standard form $a + b i$ .
${[\sqrt{3} (\cos \frac{5 蟺}{18} + i \sin \frac{5 蟺}{18})]}^{6}$

Short Answer

Expert verified

The expression in the standard form is $\frac{27}{2} - \frac{27 \sqrt{3}}{2} i$ .

Step by step solution

01

Step 1. Given information.

Consider the given question,

${[\sqrt{3} (\cos \frac{5 蟺}{18} + i \sin \frac{5 蟺}{18})]}^{6}$

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 $z = (a + i b)$ ,

$z^{6} = {[\sqrt{3} (\cos \frac{5 蟺}{18} + i \sin \frac{5 蟺}{18})]}^{6}$

02

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

Using the equation (i),

$z^{6} = {(\sqrt{3})}^{6} [\cos (6 路 \frac{5 蟺}{18}) + i \sin (6 路 \frac{5 蟺}{18})] z^{6} = 3^{3} (\cos \frac{5 蟺}{18} + i \sin \frac{5 蟺}{18}) z^{6} = 27 (\frac{1}{2} - \frac{\sqrt{3}}{2} i) z^{6} = \frac{27}{2} - \frac{27 \sqrt{3}}{2} i$

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Most popular questions from this chapter

In Problems 47鈥�52, find the unit vector in the same direction as $v$ : $v = 2 i - j$

Is the sine function even, odd, or neither?

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

$(8.1, 5.2)$

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

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

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$(6.3, 3.8)$

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