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.

$\frac{\mathbf{5}\mathbf{-}\mathbf{2}\mathbf{i}}{\mathbf{5}\mathbf{+}\mathbf{2}\mathbf{i}}$.

Short answer

The complex number can be written $z=e^{-43.6i}$.

The graph is mentioned below:

Step-by-step solution

Given Information

$\frac{5-2\mathrm{i}}{5+2\mathrm{i}}$The complex number is .

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 $\frac{5-2\mathrm{i}}{5+2\mathrm{i}}$.

Rationalize the complex number as:

$$\begin{gathered} z=\frac{5-2\mathrm{i}}{5+2\mathrm{i}}\times \frac{5-2\mathrm{i}}{5-2\mathrm{i}} \\ =\frac{25-20i+4i^2}{25-4i^2} \\ =\frac{21-20i}{29} \\ =\frac{21}{29}-\frac{20}{29}i \end{gathered}$$

Find x and y.

$$\begin{gathered} x=\frac{21}{29} \\ y=-\frac{20}{29} \end{gathered}$$

Find the value of r.

$$\begin{gathered} r=\sqrt{x^2+y^2} \\ r=\sqrt{\frac{441}{841}+\frac{400}{841}} \\ =1 \end{gathered}$$

Find the value of .

$$\begin{gathered} \theta ={\sin }^{-1}\left(\frac{y}{r}\right) \\ \theta ={\sin }^{-1}\left(-\frac{20}{29}\right) \\ =-43.6° \end{gathered}$$

The complex number can be written as $z=e^{-43.6°i}$.

The graph is mentioned below as: