Question

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

Short answer

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

Step-by-step solution

Euler’s formula:

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

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.

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

Graphical representation of the solution

The function is sketched in figure 1 as follows.

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