Chapter 9: Q. 41 (page 592)

Write the expression in the standard form $a + b i$ .
${[4 (\cos 40 ° + i \sin 40 °)]}^{3}$

Short Answer

Expert verified

The expression in the standard form is $- 32 + 32 \sqrt{3} i$ .

Step by step solution

Step 1. Given information.

Consider the given question,

${[4 (\cos 40 ° + i \sin 40 °)]}^{3}$

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 $z = (a + b i)$ ,

$z^{3} = {[4 (\cos 40 ° + i \sin 40 °)]}^{3}$

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

Using the equation (i),

$z^{3} = 4^{3} [\cos (3 \cdot 40 °) + i \sin (3 \cdot 40 °)] z^{3} = 64 [\cos 120 ° + i \sin 120 °] z^{3} = 64 (- \frac{1}{2} + i \frac{\sqrt{3}}{2}) z^{3} = - 32 + i (32 \cdot \sqrt{3})$

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Write the expression in the standard form $a + b i$ .
${[4 (\cos 40 ° + i \sin 40 °)]}^{3}$

Short Answer

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+bi.4cos40°+isin40°3

Short Answer

Step by step solution

Step 1. Given information.

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

One App. One Place for Learning.

Most popular questions from this chapter

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Write the expression in the standard form $a + b i$ .
${[4 (\cos 40 ° + i \sin 40 °)]}^{3}$