Chapter 6.6, Problem 74E

(a)

To determine

The fifth roots of $32\left(\cos \frac{5\pi }{6}+i\sin \frac{5\pi }{6}\right)$ using formula for roots.

Answer to Problem 74E

The fifth roots of $32\left(\cos \frac{5\pi }{6}+i\sin \frac{5\pi }{6}\right)$ are $\begin{array}{l}{z}_0=2\left(\cos \frac{5\pi }{30}+i\sin \frac{5\pi }{30}\right),z_1=2\left(\cos \frac{17\pi }{30}+i\sin \frac{17\pi }{30}\right),z_2=2\left(\cos \frac{29\pi }{30}+i\sin \frac{29\pi }{30}\right)\\ z_3=2\left(\cos \frac{41\pi }{30}+i\sin \frac{41\pi }{30}\right)\text{and }{z}_4=2\left(\cos \frac{53\pi }{30}+i\sin \frac{53\pi }{30}\right)\end{array}$

Explanation of Solution

Given info:

  $\sqrt[5]{32\left(\cos \frac{5\pi }{6}+i\sin \frac{5\pi }{6}\right)}$

Formula used:

The trigonometric form of the complex number $z=a+ib$ is

  $z=r\left(\cos \theta +i\sin \theta \right)$

Where $r=\sqrt{a^2+b^2}$ and $\theta ={\tan }^{-1}\left(\frac{b}{a}\right)$

For a positive integer n the complex number $z=r\left(\cos \theta +i\sin \theta \right)$ has exactly n distinct nth roots given by

  $z_k=\sqrt[n]{r}\left(\cos \left(\frac{\theta +2\pi k}{n}\right)+i\sin \left(\frac{\theta +2\pi k}{n}\right)\right)$

where k = 0, 1, 2, . . . , ,n -1.

Calculation:

We have

  $z_k=\sqrt[n]{r}\left(\cos \left(\frac{\theta +2\pi k}{n}\right)+i\sin \left(\frac{\theta +2\pi k}{n}\right)\right)$

  $n=5,r=32,\theta =\frac{5\pi }{6}$

  $z_0=\sqrt[5]{32}\left(\cos \left(\frac{\frac{5\pi }{6}+2\pi \times 0}{5}\right)+i\sin \left(\frac{\frac{5\pi }{6}+2\pi \times 0}{5}\right)\right)$

  $z_0=2\left(\cos \frac{5\pi }{30}+i\sin \frac{5\pi }{30}\right)$

  $z_1=\sqrt[5]{32}\left(\cos \left(\frac{\frac{5\pi }{6}+2\pi \times 1}{5}\right)+i\sin \left(\frac{\frac{5\pi }{6}+2\pi \times 1}{5}\right)\right)$

  $z_1=2\left(\cos \frac{17\pi }{30}+i\sin \frac{17\pi }{30}\right)$

  $z_2=\sqrt[5]{32}\left(\cos \left(\frac{\frac{5\pi }{6}+2\pi \times 2}{5}\right)+i\sin \left(\frac{\frac{5\pi }{6}+2\pi \times 2}{5}\right)\right)$

  $z_2=2\left(\cos \frac{29\pi }{30}+i\sin \frac{29\pi }{30}\right)$

  $z_3=\sqrt[5]{32}\left(\cos \left(\frac{\frac{5\pi }{6}+2\pi \times 3}{5}\right)+i\sin \left(\frac{\frac{5\pi }{6}+2\pi \times 3}{5}\right)\right)$

  $z_3=2\left(\cos \frac{41\pi }{30}+i\sin \frac{41\pi }{30}\right)$

  $z_4=\sqrt[5]{32}\left(\cos \left(\frac{\frac{5\pi }{6}+2\pi \times 4}{5}\right)+i\sin \left(\frac{\frac{5\pi }{6}+2\pi \times 4}{5}\right)\right)$

  $z_4=2\left(\cos \frac{53\pi }{30}+i\sin \frac{53\pi }{30}\right)$

The fifth roots of $32\left(\cos \frac{5\pi }{6}+i\sin \frac{5\pi }{6}\right)$ are $\begin{array}{l}{z}_0=2\left(\cos \frac{5\pi }{30}+i\sin \frac{5\pi }{30}\right),z_1=2\left(\cos \frac{17\pi }{30}+i\sin \frac{17\pi }{30}\right),z_2=2\left(\cos \frac{29\pi }{30}+i\sin \frac{29\pi }{30}\right)\\ z_3=2\left(\cos \frac{41\pi }{30}+i\sin \frac{41\pi }{30}\right)\text{and }{z}_4=2\left(\cos \frac{53\pi }{30}+i\sin \frac{53\pi }{30}\right)\end{array}$

Conclusion:

Thus,the fifth roots of $32\left(\cos \frac{5\pi }{6}+i\sin \frac{5\pi }{6}\right)$ are $\begin{array}{l}{z}_0=2\left(\cos \frac{5\pi }{30}+i\sin \frac{5\pi }{30}\right),z_1=2\left(\cos \frac{17\pi }{30}+i\sin \frac{17\pi }{30}\right),z_2=2\left(\cos \frac{29\pi }{30}+i\sin \frac{29\pi }{30}\right)\\ z_3=2\left(\cos \frac{41\pi }{30}+i\sin \frac{41\pi }{30}\right)\text{and }{z}_4=2\left(\cos \frac{53\pi }{30}+i\sin \frac{53\pi }{30}\right)\end{array}$

(b)

To determine

The standard form of each of the roots.

Answer to Problem 74E

The standard formof the roots are $\begin{array}{l}{z}_0=1.732+i,z_1=-0.416+1.956i,z_2=-1.989+0.209i,z_3=-0.813-1.827i\\ \text{and }{z}_4=1.486-1.338i\end{array}$

Explanation of Solution

Given info:

The fifth roots of $32\left(\cos \frac{5\pi }{6}+i\sin \frac{5\pi }{6}\right)$ are $\begin{array}{l}{z}_0=2\left(\cos \frac{5\pi }{30}+i\sin \frac{5\pi }{30}\right),z_1=2\left(\cos \frac{17\pi }{30}+i\sin \frac{17\pi }{30}\right),z_2=2\left(\cos \frac{29\pi }{30}+i\sin \frac{29\pi }{30}\right)\\ z_3=2\left(\cos \frac{41\pi }{30}+i\sin \frac{41\pi }{30}\right)\text{and }{z}_4=2\left(\cos \frac{53\pi }{30}+i\sin \frac{53\pi }{30}\right)\end{array}$

Formula used:

The trigonometric form of the complex number $z=a+ib$ is

  $z=r\left(\cos \theta +i\sin \theta \right)$

Where $r=\sqrt{a^2+b^2}$ and $\theta ={\tan }^{-1}\left(\frac{b}{a}\right)$

Calculation:

The roots from part a are $\begin{array}{l}{z}_0=2\left(\cos \frac{5\pi }{30}+i\sin \frac{5\pi }{30}\right),z_1=2\left(\cos \frac{17\pi }{30}+i\sin \frac{17\pi }{30}\right),z_2=2\left(\cos \frac{29\pi }{30}+i\sin \frac{29\pi }{30}\right)\\ z_3=2\left(\cos \frac{41\pi }{30}+i\sin \frac{41\pi }{30}\right)\text{and }{z}_4=2\left(\cos \frac{53\pi }{30}+i\sin \frac{53\pi }{30}\right)\end{array}$

We have

  $\begin{array}{l}{z}_0=2\left(\cos \frac{5\pi }{30}+i\sin \frac{5\pi }{30}\right)=1.732+i\\ z_1=2\left(\cos \frac{17\pi }{30}+i\sin \frac{17\pi }{30}\right)=-0.416+1.956i\\ z_2=2\left(\cos \frac{29\pi }{30}+i\sin \frac{29\pi }{30}\right)=-1.989+0.209i\\ z_3=2\left(\cos \frac{41\pi }{30}+i\sin \frac{41\pi }{30}\right)=-0.813-1.827i\\ z_4=2\left(\cos \frac{53\pi }{30}+i\sin \frac{53\pi }{30}\right)=1.486-1.338i\end{array}$

The standard formof the roots are

  $\begin{array}{l}{z}_0=1.732+i,z_1=-0.416+1.956i,z_2=-1.989+0.209i,z_3=-0.813-1.827i\\ \text{and }{z}_4=1.486-1.338i\end{array}$

Conclusion:

Thus,the standard formof the roots are $\begin{array}{l}{z}_0=1.732+i,z_1=-0.416+1.956i,z_2=-1.989+0.209i,z_3=-0.813-1.827i\\ \text{and }{z}_4=1.486-1.338i\end{array}$ .

(c)

To determine

The roots graphically.

Answer to Problem 74E

Please refer calculation part.

Explanation of Solution

Given info:

The fifth roots of $32\left(\cos \frac{5\pi }{6}+i\sin \frac{5\pi }{6}\right)$ are $\begin{array}{l}{z}_0=1.732+i,z_1=-0.416+1.956i,z_2=-1.989+0.209i,z_3=-0.813-1.827i\\ \text{and }{z}_4=1.486-1.338i\end{array}$

Formula used:

The trigonometric form of the complex number $z=a+ib$ is

  $z=r\left(\cos \theta +i\sin \theta \right)$

Where $r=\sqrt{a^2+b^2}$ and $\theta ={\tan }^{-1}\left(\frac{b}{a}\right)$

Calculation:

Graphical representation of roots are given by

  

Conclusion:

Thus,the graphical representation of roots are given.