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.

$\left(cos\mathit{\pi }\mathit{-}\mathit{i}\mathit{}\sin \mathit{\pi }\right)$.

Short answer

The required values are mentioned below:

$x=-1,y=0,r=1,\theta =-\pi$

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

Step-by-step solution

Given Information

The complex number is $(\cos \pi -i\sin \pi )$.

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

$\begin{array}{l}x=c\mathrm{os}(-\mathrm{\pi })\\ =-1\\ y=sin(-\mathrm{\pi })\\ =0\end{array}$

Find the value of r:

Compare with the value of x.

localid="1658476128766" $\begin{array}{l}x=cos(-\pi )\\ r=1\end{array}$

Find the value of $\theta$.

Compare with the value of x.

localid="1658476133946" $\begin{array}{l}x=cos(-\pi )\\ \theta =-\pi \end{array}$

The value of the number becomes as follows.

localid="1658476145457" $\cos \left(\pi \right)-i\sin \left(\pi \right)e^{i\theta }=\left(e^{-i\pi }\right)$

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

The required values are mentioned below:

$x=-1,y=0,r-=1,\theta =\pi$