Q12P - 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.

$\mathbf{3}\mathbf{(}\mathit{c}\mathit{o}\mathit{s}\mathbf{28}\mathbf{°}\mathbf{+}\mathit{i}\mathit{s}\mathit{i}\mathit{n}\mathbf{28}\mathbf{°}\mathbf{)}$ .

Step-by-Step Solution

Answer

The complex number can be written as $z=3e^{0.49i}$ .

 

The graph is mentioned below:

Step 1: Given Information

The complex number is $3(cos28°+isin28°)$.

Step 2: Definition of the Complex number.

A complex number is a number that consists of real and imaginary numbers and is written in the form as:

$\mathit{z}\mathbf{=}\mathit{a}\mathbf{+}\mathit{i}\mathit{b}$ where a and b are real, and z is the complex number.

Step 3: Find the values

The complex number is $3(cos28°+isin28°)$ .

$$\begin{gathered} z=\theta (cos\theta +sin\theta ) \\ r=3 \\ \theta =28° \end{gathered}$$ 

 

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:

 $$\begin{gathered} x=3 \cos (28°) \\ =2.64 \\ y=3\sin (28°) \\ =1.41 \\ z=3e^{0.49i} \end{gathered}$$

 

The complex number can be written as $z=3e^{0.49i}$.

 

The graph is mentioned below: