Q14P - Mathematical Methods in Physical Sciences

Question

For each of the following numbers, first visualize where it is in the complex plane. With a little practice you can quickly find $\mathit{x}\mathbf{,}\mathit{y}\mathbf{,}\mathit{r}\mathbf{,}\mathit{\theta }$ in your head for these simple problems. Then plot the number and label it in five ways as in Figure 3.3. Also plot the complex conjugate of the number.

  $\mathbf{2}\left(cos\frac{\pi }{4}+i sin\frac{\pi }{4}\right)$. 

Step-by-Step Solution

Answer

The required values are mentioned below:

$x=\sqrt{2},y=\sqrt{2},r=2,\theta =\frac{\mathrm{\pi }}{4}$ 

 

The graph of the number and its conjugate is shown below:

 

Step 1: Given Information

The complex number is  $2\left(cos\frac{\pi }{4}+isin\frac{\pi }{4}\right)$.

Step 2: Definition of the Complex number

Real numbers can be written as complex numbers with the help of iota (i).

$\mathit{z}\mathbf{=}\mathit{a}\mathbf{+}\mathit{\tau }\mathit{b}$ where a and b are real numbers.

Step 3: Find the value

The formula is mentioned below:

$$\begin{gathered} x+iy=r\left(\cos \theta +i\sin \theta \right) \\ =r{e}^{ie} \end{gathered}$$ 

 

The formulas for x and y are given below:

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

 

Find x and y as:

$$\begin{gathered} x=2\cos \left(\frac{\pi }{4}\right) \\ =\sqrt{2} \\ y=2\sin \left(\frac{\pi }{4}\right) \\ =\sqrt{2} \end{gathered}$$ 

 

Find the value of  r :

Compare with the value of x.

$\begin{array}{l}x=2cos\left(\frac{\pi }{4}\right)\\ r=2\end{array}$ 

 

Find the value of  $\theta$.

Compare with the value of x.

$$\begin{gathered} x=2\cos \left(\frac{\pi }{4}\right) \\ \theta =\frac{\pi }{4} \end{gathered}$$ 

 

The value of the number becomes as follows.

$2\left(\cos \left(\frac{\pi }{4}\right)+i \sin \left(\frac{\pi }{4}\right)\right)r{e}^{ie}=2\left(e^{i\pi /4}\right)$ 

 

The graph of the complex number (blue) and its conjugate (red)  is shown below:

 

The required values are mentioned below:

 $x=\sqrt{2},y=\sqrt{2},r=2,\theta =\frac{\pi }{4}$