Question

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

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

Short answer

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

The graph is shown below as:

Step-by-step solution

Given Information

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

Definition of the Complex number

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

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

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:

localid="1658486574125" $$\begin{gathered} \theta =-40° \\ =-\frac{2\mathrm{\pi }}{9} \\ r=2 \end{gathered}$$

The formulas for x and y are given below:

localid="1658486578027" $$\begin{gathered} x=r\cos \left(\theta \right) \\ y=r\sin \left(\theta \right) \end{gathered}$$

Find x and y

localid="1658486580877" $$\begin{gathered} x=2\cos (-40°) \\ =1.53 \\ y=2\sin (-40°) \\ =-1.28 \\ z=1.53-0.28i \end{gathered}$$

The complex number becomes localid="1658486595837" $z=1.53-0.28i$.

The graph is shown below: