Q16P - Mathematical Methods in Physical Sciences

Question

Question:First simplify each of the following numbers to the x+iy form or to the $\mathit{r}{\mathit{e}}^{\mathbf{i}\mathbf{\theta }}$ form. Then plot the number in the complex plane.

$\frac{\mathbf{1}}{\mathbf{0}\mathbf{.}\mathbf{5}\mathbf{(}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{40}\mathbf{°}\mathbf{+}\mathbf{i}\mathbf{ }\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{40}\mathbf{°}}$ .

Step-by-Step Solution

Answer

The complex number can be written as z = 1.53 - 1.28i .

 

The graph is shown below as:

Step 1: Given Information

The complex number is $\frac{1}{0.5(\cos 40°+\mathrm{i} \sin 40°}$.

Step 2: Definition of the Complex number

Real and imaginary numbers combine and form a complex number as:

z = a + ib Here a and b are the real numbers, and z is a complex number

Step 3: Find the values

The complex number is  .

 $\frac{1}{0.5(\cos 40°+\mathrm{i} \sin 40°}$

Factorize the complex number as:

 $$\begin{gathered} 0.5\left(\cos \right(40°)+i \sin (40°\left)\right)=0.5e^{i(2\pi /9)} \\ \frac{1}{0.5\left(\cos \right(40°)+i \sin (40°\left)\right)}=\frac{1}{0.5e^{i(2\pi /9)}} \\ =2e^{i(2\pi /9)} \end{gathered}$$

 

The value of r and $\theta$ are given below:

 $$\begin{gathered} \theta =-40° \\ =-\frac{2\mathrm{\pi }}{9} \\ r=2 \end{gathered}$$

 

The formulas for x and y are given below:

 $$\begin{gathered} x=r \cos \left(\theta \right) \\ y=r \sin \left(\theta \right) \end{gathered}$$

 

Find x and y

$$\begin{gathered} x=2 \cos (-40°) \\ =1.53 \\ y=2 \sin (-40°) \\ =-1.28 \\ z=1.53-0.28i \end{gathered}$$ 

 

The complex number becomes $z=1.53-0.28i$ .

 

The graph is shown below: