Chapter 6.6, Problem 96E

(a)

To determine

To Find: The number of roots that is not shown.

Answer to Problem 96E

The number of roots not shown is 3.

Explanation of Solution

Given:

The given Figure is shown in Figure 1

  

Figure 1

Calculation:

The figure shows one of the fourth root and only one is shown in the given figure.

Thus, the number of roots not shown is 3.

(b)

To determine

To Find: The roots of the complex number.

Answer to Problem 96E

The roots are $2\left(\cos 97.5°i+\sin 97.5°\right)$ , $2\left(\cos 187.5°i+\sin 187.5°\right)$ and

  $2\left(\cos 277.5°i+\sin 277.5°\right)$ .

Explanation of Solution

The figure shows that the number of roots are $\frac{360°}{n}$ apart for the number of roots and being 4. And the remaining roots lie on the circle. To find the other roots start from $90°$ to other equation and add continuously to find the other roots.

The fourth root of the complex number is $z=r\left(\cos \frac{\theta +2\pi k}{4}+i\sin \frac{\theta +2\pi k}{4}\right)$ .

Consider one of the fourth root of the complex number is $z_0=2\left(\cos 30°+i\sin 30°\right)$ , here $\theta =30°$ and $\sqrt[4]{r}=2$ .

The first root is calculated as,

  $\begin{array}{c}{z}_1=\sqrt[4]{r}\left(\cos \frac{\theta +2\pi }{4}+i\sin \frac{\theta +2\pi }{4}\right)\\ =2\left(\cos 97.5°i+\sin 97.5°\right)\end{array}$

The second root is calculated as,

  $\begin{array}{c}{z}_2=\sqrt[4]{r}\left(\cos \frac{\theta +2\pi \left(2\right)}{4}+i\sin \frac{\theta +2\pi \left(2\right)}{4}\right)\\ =2\left(\cos 187.5°i+\sin 187.5°\right)\end{array}$

The third root is calculated as,

  $\begin{array}{c}{z}_2=\sqrt[4]{r}\left(\cos \frac{\theta +2\pi \left(3\right)}{4}+i\sin \frac{\theta +2\pi \left(3\right)}{4}\right)\\ =2\left(\cos 277.5°i+\sin 277.5°\right)\end{array}$

Thus, the roots are $2\left(\cos 97.5°i+\sin 97.5°\right)$ , $2\left(\cos 187.5°i+\sin 187.5°\right)$ and

  $2\left(\cos 277.5°i+\sin 277.5°\right)$ .