Q20P - 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.

$\mathbf{7}\mathbf{(}\mathit{c}\mathit{o}\mathit{s}{\mathbf{110}}^{\mathbf{°}}\mathbf{-}\mathit{i}\mathit{s}\mathit{i}\mathit{n}{\mathbf{110}}^{\mathbf{°}}\mathbf{)}$ .

Step-by-Step Solution

Answer

The required values are mentioned below:

 $x=-2.4,y=-6.67,r=7,\theta =-{110}^{°}$

 

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

Step 1: Given Information

The complex number is $7(cos{110}^{°}-isin{110}^{°})$ .

Step 2: Definition of the Complex number

Complex numbers have two types of numbers in them:

(a) Real number

(b) Imaginary number

 

  $\mathit{z}\mathbf{=}\mathit{a}\mathbf{+}{}_{\mathbf{\iota }}\mathit{b}$z is a complex number, and a and b are real numbers.

Step 3: Find the value

The formula is mentioned below:

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

 

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:

 $$\begin{gathered} x=7\cos \left({110}^{°}\right) \\ =-2.4 \\ y=-7\sin \left({110}^{°}\right) \\ =-6.67 \end{gathered}$$

 

Find the value of r :

Compare with the value of x.

$$\begin{gathered} x=7\cos \left({110}^{°}\right) \\ r=7 \end{gathered}$$ 

 

Find the value of $\theta$ .

Compare with the value of x.

$$\begin{gathered} x=7\cos \left({110}^{°}\right) \\ \theta =-{110}^{°} \end{gathered}$$ 

 

The value of the number becomes as follows:

 $\left(\cos \left({110}^{°}\right)-i\sin \left({110}^{°}\right)\right)=7\left(e^{-i(11\mathrm{\pi }/18)}\right)$

 

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

The required values are mentioned below:

$x=-2.4,y=-6.67,r=7,\theta =-{110}^{°}$