Chapter 9: Q. 60 (page 592)

In Problems 53–60, find all the complex roots. Leave your answers in polar form with the argument in degrees.
60. The complex fifth roots of $- i$ .

Short Answer

Expert verified

The complex fifth roots of $- i$ are $\cos 54 + i \sin 54, \cos 126 + i \sin 126, \cos 198 + i \sin 198, \cos 270 + i \sin 270, \cos 342 + \sin 342$ .

Step by step solution

Step 1. Rewrite -i in the polar form.

The magnitude of $- i$ is

$\sqrt{0 + {(- 1)}^{2}} = \sqrt{1} = 1$

Which gives

$- i = 1 (0 - i) = (\cos 270 ° - i \sin 270 °)$

Step 2. Using De Moivre's Theorem.

We get

$z_{k} = \sqrt[5]{1} [\cos (\frac{270}{5} + \frac{360 k}{5}) + i \sin (\frac{270}{5} + \frac{360 k}{5})] = [\cos (54 ° + 72 ° k) + i \sin (54 ° + 72 ° k)]; k = 0, 1, 2, 3, 4$

Step 3. Substitute k=0,1,2,3,4 in zk=cos(54+72k)+isin(54+72k).

We get

$z_{0} = \cos 54 + i \sin 54 z_{1} = \cos 126 + i \sin 126 z_{2} = \cos 198 + i \sin 198 z_{3} = \cos 270 + i \sin 270 z_{4} = \cos 342 + i \sin 342$

So we have found the complex fifth roots of $- i$ .

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In Problems 53–60, find all the complex roots. Leave your answers in polar form with the argument in degrees.
60. The complex fifth roots of $- i$ .

Short Answer

Step by step solution

Step 1. Rewrite -i in the polar form.

Step 2. Using De Moivre's Theorem.

Step 3. Substitute k=0,1,2,3,4 in zk=cos(54+72k)+isin(54+72k).

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In Problems 53–60, find all the complex roots. Leave your answers in polar form with the argument in degrees.60. The complex fifth roots of-i.

Short Answer

Step by step solution

Step 1. Rewrite -i in the polar form.

Step 2. Using De Moivre's Theorem.

Step 3. Substitute k=0,1,2,3,4 in zk=cos(54+72k)+isin(54+72k).

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Theoretical and Mathematical Physics

Pure Maths

Probability and Statistics

Geometry

Applied Mathematics

Study anywhere. Anytime. Across all devices.

In Problems 53–60, find all the complex roots. Leave your answers in polar form with the argument in degrees.
60. The complex fifth roots of $- i$ .