Q11P - 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.

 $\mathbf{2}\mathbf{(}\mathbf{cos}\frac{\mathbf{\pi }}{\mathbf{6}}\mathbf{+}\mathbf{i}\mathbf{ }\mathbf{sin}\frac{\mathbf{\pi }}{\mathbf{6}}\mathbf{)}$. 

Step-by-Step Solution

Answer

The required values are mentioned below:

 $x=\sqrt{3},y=1,r=2,\theta =\frac{\pi }{6}$

 

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

Step 1: Given Information

The complex number is $2(cos\frac{\mathit{\pi }}{\mathit{6}}\mathit{+}i\mathit{ }sin\frac{\mathit{\pi }}{\mathit{6}})$.

Step 2: Definition of the Complex number

A complex number is a number that can be represented as:

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

Step 3: 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:

$$\begin{gathered} x=2\cos \left(\frac{\pi }{6}\right) \\ =\sqrt{3} \\ y=2\sin \left(\frac{\pi }{6}\right) \\ =1 \end{gathered}$$

Find the value of  .

Compare with the value of x, and we get,

 $$\begin{gathered} r \cos \theta =2 \cos \left(\frac{\pi }{6}\right) \\ r=2 \end{gathered}$$

 

Find the value of $\theta$ .

Compare with the value of x, and we get,

 $$\begin{gathered} r \cos \theta =2 \cos \left(\frac{\pi }{6}\right) \\ \mathrm{\theta }=\frac{\pi }{6} \end{gathered}$$

 

The value of the number becomes as follows:

 $2\left(\cos \left(\frac{\pi }{6}\right)+i \sin \left(\frac{\pi }{6}\right)=2\left(e^{i\pi /6}\right)\right)$

 

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

 

The required values are mentioned below:

$x=\sqrt{3},y=1,r=2,\theta =\frac{\pi }{6}$