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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. 45 (page 592)

Write the expression in the standard form $a + b i$ .
${[\sqrt{3} (\cos 10 掳 + i \sin 80 掳)]}^{3}$

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,

${[\sqrt{3} (\cos 10 掳 + i \sin 80 掳)]}^{3}$

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 $z = (a + i b)$ ,

$x^{6} = {[\sqrt{3} (\cos 10 掳 + i \sin 10 掳)]}^{6}$

02

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

Using the equation (i),

$z^{6} = {(\sqrt{3})}^{6} [\cos (6 路 10 掳) + i \sin (6 路 10 掳)] z^{6} = 3^{3} (\cos 60 掳 + i \sin 60 掳) z^{6} = 27 (\frac{1}{2} + \frac{\sqrt{3}}{2} i) z^{6} = \frac{27}{2} + \frac{27 \sqrt{3}}{2} i$

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Most popular questions from this chapter

In Problem, the vector v has initial point P and terminal point Q. Write v in the form ai + bj; that is, find its position vector.

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

$r = 1 - 3 \cos 胃$

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