Q13P - Mathematical Methods in Physical Sciences

Question

First simplify each of the following numbers to the$\mathit{x}\mathbf{+}\mathit{i}\mathit{y}$   form or to the   form. Then plot the number in the complex plane.  $\mathbf{5}\mathbf{(}\mathbf{c}\mathbf{o}\mathbf{s}\frac{\mathbf{2}\mathbf{\pi }}{\mathbf{5}}\mathbf{+}\mathbf{i}\mathbf{s}\mathbf{i}\mathbf{n}\frac{\mathbf{2}\mathbf{\pi }}{\mathbf{5}}\mathbf{)}$

 .

Step-by-Step Solution

Answer

The complex number becomes .$5e^{\frac{2\pi }{5}i}$

Step 1: Given Information

The complex number is.$5(\cos \frac{2\pi }{5}+\sin \frac{2\pi }{5})$

Step 2: Definition of the Complex number

A complex number is a number that is made up of both real and imaginary numbers.

Step 3: Find the value

The complex number is.$5(\cos \frac{2\pi }{5}+\sin \frac{2\pi }{5})$

 $\begin{array}{l}z=r(\cos \theta +\sin \theta )\\ r=5\\ \theta =\frac{2\pi }{5}\end{array}$

 

The complex number in the form of$r{e}^{i\theta }$ is,

 $\begin{array}{c}z=r{e}^{i\theta }\\ =5e^{\frac{2\pi }{5}i}\end{array}$

Hence, the complex number becomes .$5e^{\frac{2\pi }{5}i}$

 

The graph is mentioned below.