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Found in: Page 437

### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

# Sketch the trajectory of the complex-valued function ${\mathbf{z}}{\mathbf{=}}{{\mathbf{e}}}^{\mathbf{3}\mathbf{it}}$and find its period.

The period is $\frac{2\tau \tau }{3}$.

See the step by step solution

## Step1: Euler’s formula:

The Euler’s formula is ${{\mathbit{e}}}^{\mathbf{i}\mathbf{t}}{\mathbf{=}}{\mathbit{c}}{\mathbit{o}}{\mathbit{s}}{\mathbf{}}{\mathbit{t}}{\mathbf{+}}{\mathbit{i}}{\mathbf{}}{\mathbit{s}}{\mathbit{i}}{\mathbit{n}}{\mathbf{}}{\mathbit{t}}$

## Step2: Explanation of the solution

Consider the given complex valued function:

$z={e}^{3it}$

By using Euler’s formula, the function can be written as follows.

$\begin{array}{l}\mathrm{z}={\mathrm{e}}^{3\mathrm{it}}\\ \mathrm{Z}=\mathrm{cos}\left(3\mathrm{t}\right)+\mathrm{isin}\left(3\mathrm{t}\right)\end{array}$

## Step3: Graphical representation of the solution

The function is sketched in figure 1 as follows.

Hence, the period for the function $z={e}^{3it}$ is $\frac{2\tau \tau }{3}$.