Q18P - 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 $\mathit{r}{\mathit{e}}^{\mathbf{i}\mathbf{\theta }}$   form. Then plot the number in the complex plane.

${\mathbf{(}\mathbf{0}\mathbf{.64}\mathbf{+}\mathbf{0}\mathbf{.77}\mathbf{i}\mathbf{)}}^{\mathbf{4}}$ .

Step-by-Step Solution

Answer

The complex number becomes .$z=-0.93-0.35i$

 

The graph is mentioned below.

Step 1: Given Information

Thecomplex number is ${(0.64+0.77i)}^4$

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 ${(0.64+0.77i)}^4$

Find the values for  $(0.64+0.77i)$

Find x and y.

 $\begin{array}{c}z=(0.64+0.77i)\\ x=0.64\\ y=0.77\end{array}$

 

Find the value of  r

$\begin{array}{c}r=\sqrt{x^2+y^2}\\ r=\sqrt{x{\left(0.64\right)}^2+{\left(0.77\right)}^2}\\ =1\end{array}$

Find the value of .$\theta$

 $\begin{array}{c}\theta ={\cos }^{-1}\left(\frac{x}{r}\right)\\ \theta ={\cos }^{-1}\left(\frac{0.64}{1}\right)\\ =0.876\text{rad}\end{array}$

$0.64+0.77i=e^{i\left(0.876\right)}$

For the complex number  .${(0.64+0.77i)}^4$

$\begin{array}{c}z\text{t}={\left(0.64+0.77i\right)}^4\\ ={\left(e^{i\left(0.876\right)}\right)}^4\\ =\left(e^{i(4\times 0.876)}\right)\\ \left(e^{i(4\times 0.876)}\right)=e^{i\left(3.5\right)}\end{array}$

Find x and y.

 $\begin{array}{c}x=\cos \left(3.5\text{rad}\right)\\ =-0.93\\ y=\sin \left(3.5\text{rad}\right)\\ =-0.35\end{array}$

 

The complex number becomes  $z=-0.93-0.35i$

The graph is mentioned below.

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