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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. 46 (page 592)

Write the expression in the standard form $a + i b$ .
${[\frac{1}{2} (\cos 72 掳 + i \sin 72 掳)]}^{5}$

Short Answer

Expert verified

The expression in the standard form is $\frac{1}{32}$ .

Step by step solution

01

Step 1. Given information.

Consider the given question,

${[\frac{1}{2} (\cos 72 掳 + i \sin 72 掳)]}^{5}$

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 $z = (a + b i)$ ,

$z^{5} = {[\frac{1}{2} (\cos 72 掳 + i \sin 72 掳)]}^{5}$

02

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

Using the equation (i),

$z^{5} = {(\frac{1}{2})}^{5} [\cos (5 路 72 掳) + i \sin (5 路 72 掳)] z^{5} = \frac{1}{32} (\cos 360 掳 + i \sin 360 掳) z^{5} = \frac{1}{32} (1 + i 路 0) z^{5} = \frac{1}{32}$

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Most popular questions from this chapter

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

${[4 (\cos 40 掳 + i \sin 40 掳)]}^{3}$

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

$r = 4 \cos (3 胃)$

To test if the graph of a polar equation may be symmetric with respect to the polar axis, replace $胃$ by ________.

If the rectangular coordinates of a point are $(4, - 6),$ the point symmetric to it with respect to the origin is ______.

Identify and graph each polar equation. Verify your graph using a graphing utility.
$r = 1 - 2 \sin 胃$

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