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.

$\mathbf{4}\mathbf{(}\mathbf{cos}\frac{\mathbf{2}\mathbf{\pi }}{\mathbf{3}}\mathbf{+}\mathbf{i}\mathbf{}\mathbf{sin}\frac{\mathbf{2}\mathbf{\pi }}{\mathbf{3}}\mathbf{)}$.

Short answer

The required values are mentioned below:

$x=2,y=-2\sqrt{3},r=4,\theta =-\frac{2\pi }{3}$

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

Step-by-step solution

Given Information 

The complex number is $\mathit{4}\mathit{(}\mathit{c}\mathit{o}\mathit{s}\frac{\mathit{2}\mathit{\pi }}{\mathit{3}}\mathit{+}\mathit{i}\mathit{}\mathit{s}\mathit{i}\mathit{n}\frac{\mathit{2}\mathit{\pi }}{\mathit{3}}\mathit{)}$.

Definition of the Complex number

A complex number is made up of real numbers and imaginary numbers.

Find the value

The formula is mentioned below:

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

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 as:

$$\begin{gathered} x=4\cos \left(\frac{2\pi }{3}\right) \\ =-2 \\ y=4\sin \left(-\frac{2\pi }{3}\right) \\ =-2\sqrt{3} \end{gathered}$$

Find the value of r .

Compare the value of x:

$$\begin{gathered} x=4\cos \left(\frac{2\pi }{3}\right) \\ r=4 \end{gathered}$$

Find the value of $\theta$.

Compare the value of :

$$\begin{gathered} x=4\cos \left(\frac{2\pi }{3}\right) \\ \theta =-\frac{2\pi }{3} \end{gathered}$$

The value of the number becomes as follows:

localid="1658476518733" $4\left(\cos \left(\frac{2\pi }{3}\right)-i\sin \left(\frac{2\pi }{3}\right)=4\left(e^{-i\left(2\pi /3\right)}\right)\right)$

The required values are mentioned below:

localid="1658476523136" $x=-2,y-2\sqrt{3},r=4,\theta =-\frac{2\pi }{3}$

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