Short Answer

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

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

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information.

Consider the given question,

${[3 (\cos 80 ° + i \sin 80 °)]}^{3}$

We know that for a complex number raise to n when it is in the polar form,

$z^{n} = r^{n} (\cos θ + i \sin (n θ)) . . . . . . (i)$

Write the expression in the standard form $z = (a + b i)$ ,

$z^{3} = {[3 (\cos 80 ° + i \sin 80 °)]}^{3}$

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

Using the equation (i),

$z^{3} = 3^{3} [\cos (3 \cdot 80 °) + i \sin (3 \cdot 80 °)] z^{3} = 27 (\cos 240 ° + i \sin 240 °) z^{3} = 27 [- \frac{1}{2} + i (\frac{- \sqrt{3}}{2})] z^{3} = - \frac{27}{2} - \frac{27 \sqrt{3}}{2} i$

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Precalculus Enhanced with Graphing Utilities

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Short Answer

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

Step by Step Solution

Step 1. Given information.

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

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Step by Step Solution

Step 1. Given information.

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

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