Answered: Use DeMoivre’s Theorem to find all the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Use DeMoivre’s Theorem to find all the cube roots of 2i. Show your work.

Expert Solution

Step 1

From the given $z = 0 + 2 i$ . That is, $a = 1 a n d b = 2$ .

Recall the DeMoivre’s Theorem:

"If $Z = [r (\cos θ + i \sin θ)]$ , then the n distinct complex numbers will be $r^{\frac{1}{n}} [(\cos (\frac{θ + 2 πk}{n}) + i \sin (\frac{θ + 2 πk}{n}))]$ , $k = 0, 1, 2, \dots n - 1$ , where $r = \sqrt{a^{2} + b^{2}}$ and $θ = \tan^{- 1} (\frac{b}{a})$ ."

Compute the value of r as follows.

Substitute $a = 1 a n d b = 2$ in $r = \sqrt{a^{2} + b^{2}}$ .

$r = \sqrt{0^{2} + 2^{2}} = \sqrt{4} = 2$

Compute the value of $θ$ as follows.

$θ = \tan^{- 1} (\frac{2}{0}) θ = \tan^{- 1} (u n d e f i n e d) θ = \frac{π}{2}$

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