Chapter 9: Q. 59 (page 592)

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

Short Answer

Expert verified

The complex fifth roots of $i$ are $(\cos 18 + i \sin 18), (\cos 90 + i \sin 90), (\cos 162 + i \sin 162), (\cos 234 + i \sin 234), (\cos 306 + i \sin 306)$ .

Step by step solution

Step 1. Rewrite i in the polar form.

The magnitude of $i = 0 + i$ is

$\sqrt{0^{2} + 1^{2}} = \sqrt{1} = 1$

which gives

$i = 1 (0 + i) = 0 + i = \cos 90 ° + i \sin 90 °$

Step 2. Using the theorem of complex roots.

We get

$z_{k} = [\cos (\frac{90}{5} + \frac{360 k}{5}) + \sin (\frac{90}{5} + \frac{360 k}{5}) i]; k = 0, 1, 2, 3, 4 = [\cos (18 ° + 72 ° k) + i \sin (18 ° + 72 ° k)]$

Step 3. Substitute k=0,1,2,3,4 in zk=[cos(18+72k)+isin(18+72k)].

We get

$z_{0} = \cos 18 + i \sin 18 z_{1} = \cos (18 + 72) + i \sin (18 + 72) = \cos 90 + i \sin 90 z_{2} = \cos (18 + 144) + i \sin (18 + 144) = \cos 162 + i \sin 162 z_{3} = \cos (18 + 216) + i \sin (18 + 216) = \cos 234 + i \sin 234 z_{4} = \cos (18 + 288) + i \sin (18 + 288)] = \cos 306 + i \sin 306$

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.
59. The complex fifth roots of $i$ .

Short Answer

Step by step solution

Step 1. Rewrite i in the polar form.

Step 2. Using the theorem of complex roots.

Step 3. Substitute k=0,1,2,3,4 in zk=[cos(18+72k)+isin(18+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.59. The complex fifth roots ofi.

Short Answer

Step by step solution

Step 1. Rewrite i in the polar form.

Step 2. Using the theorem of complex roots.

Step 3. Substitute k=0,1,2,3,4 in zk=[cos(18+72k)+isin(18+72k)].

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Applied Mathematics

Mechanics Maths

Logic and Functions

Decision Maths

Theoretical and Mathematical Physics

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.
59. The complex fifth roots of $i$ .