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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. 47 (page 592)

Write the expression in the standard form $a + i b$ .
${[5 (\cos \frac{3 蟺}{16} + i \sin \frac{3 蟺}{16})]}^{4}$

Short Answer

Expert verified

The expression in the standard form is $- \frac{25 \sqrt{2}}{2} + \frac{25 \sqrt{2}}{2} i$ .

Step by step solution

01

Step 1. Given information.

Consider the given question,

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

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 $z = (a + i b)$ ,

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

02

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

Using the equation (i),

$z^{4} = {(\sqrt{5})}^{4} [\cos (4 路 \frac{3 蟺}{16}) + i \sin (4 路 \frac{3 蟺}{16})] z^{4} = 5^{2} (\cos \frac{3 蟺}{4} + i \sin \frac{3 蟺}{4}) z^{4} = 25 (- \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} i) z^{4} = - \frac{25 \sqrt{2}}{2} + \frac{25 \sqrt{2}}{2} i$

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Most popular questions from this chapter

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

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