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.

role="math" localid="1658473180216" $\mathbf{2}\left(cos\frac{\pi }{4}+isin\frac{\pi }{4}\right)$.

Short 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-by-step solution

Given Information

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

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.

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:

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