Chapter 6.6, Problem 110E

(a)

To determine

To calculate:

The fifth roots of complex number

Answer to Problem 110E

The fifth roots of complex number are $2\left(\cos {30}^{\circ }+i\sin {30}^{\circ }\right)$ $,2\left(\cos {102}^{\circ }+i\sin {102}^{\circ }\right)$ $,2\left(\cos {174}^{\circ }+i\sin {174}^{\circ }\right)$ $,2\left(\cos {246}^{\circ }+i\sin {246}^{\circ }\right)$ and $2\left(\cos {318}^{\circ }+i\sin {318}^{\circ }\right)$ .

Explanation of Solution

Given information:

Complex number is $32\left(\cos \frac{5\pi }{6}+i\sin \frac{5\pi }{6}\right)$

Formula used:

The formula to calculate the roots of complex number are given by,

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

Where, $k=0,1,\mathrm{2...},n-1$

Calculation:

Now first find the roots of the complex number by using the above formula,

For $k=0,1,2,3,4$

First find for $k=0$ ,

  $2\left[\cos \left(\frac{\pi }{6}\right)+i\sin \left(\frac{\pi }{6}\right)\right]$

  $=2\left(\cos {30}^{\circ }+i\sin {30}^{\circ }\right)$

Now, for $k=1$

  $\begin{array}{l}2\left[\cos \frac{1}{5}\left(\frac{5\pi }{6}+2\pi \right)+i\sin \frac{1}{5}\left(\frac{5\pi }{6}+2\pi \right)\right]\\ =2\left(\cos {102}^{\circ }+i\sin {102}^{\circ }\right)\end{array}$

For $k=2$ ,

  $\begin{array}{l}2\left[\cos \frac{1}{5}\left(\frac{5\pi }{6}+4\pi \right)+i\sin \frac{1}{5}\left(\frac{5\pi }{6}+4\pi \right)\right]\\ =2\left(\cos {174}^{\circ }+i\sin {174}^{\circ }\right)\end{array}$

For $k=3$ ,

  $\begin{array}{l}2\left[\cos \frac{1}{5}\left(\frac{5\pi }{6}+6\pi \right)+i\sin \frac{1}{5}\left(\frac{5\pi }{6}+6\pi \right)\right]\\ =2\left(\cos {246}^{\circ }+i\sin {246}^{\circ }\right)\end{array}$

Now in the last step find for $k=4$ ,

  $\begin{array}{l}2\left[\cos \frac{1}{5}\left(\frac{5\pi }{6}+8\pi \right)+i\sin \frac{1}{5}\left(\frac{5\pi }{6}+8\pi \right)\right]\\ =2\left(\cos {318}^{\circ }+i\sin {318}^{\circ }\right)\end{array}$

(b)

To determine

To calculate:

And write the roots in standard form

Answer to Problem 110E

The roots in standard form are

Explanation of Solution

Given information:

The roots of complex number are:

  $2\left(\cos {30}^{\circ }+i\sin {30}^{\circ }\right)$

  $2\left(\cos {102}^{\circ }+i\sin {102}^{\circ }\right)$

  $2\left(\cos {174}^{\circ }+i\sin {174}^{\circ }\right)$

  $2\left(\cos {246}^{\circ }+i\sin {246}^{\circ }\right)$

  $2\left(\cos {318}^{\circ }+i\sin {318}^{\circ }\right)$

Calculation:

Now here to find the standard form of the roots,

As the value of $\cos {30}^{\circ }=\frac{\sqrt{3}}{2}$

And $\sin {30}^{\circ }=\frac{1}{2}$

Now write the standard form of all the given roots in trigonometric form.

For $2\left(\cos {30}^{\circ }+i\sin {30}^{\circ }\right)$

  $=\sqrt{3}+i$

The standard form of $2\left(\cos {102}^{\circ }+i\sin {102}^{\circ }\right)$

  $=-0.416+1.956i$

The standard form of $2\left(\cos {174}^{\circ }+i\sin {174}^{\circ }\right)$

  $=-1.989+0.209i$

The standard form of

  $\begin{array}{l}2\left(\cos {246}^{\circ }+i\sin {246}^{\circ }\right)\\ =-0.8134-1.827i\end{array}$

Now the standard form of $2\left(\cos {318}^{\circ }+i\sin {318}^{\circ }\right)$

  $=-1.4862-1.3382i$

(c)

To determine

To graph:

The roots in graphical form.

Explanation of Solution

Given information:

Roots given,

  $2\left(\cos {30}^{\circ }+i\sin {30}^{\circ }\right)$

  $2\left(\cos {102}^{\circ }+i\sin {102}^{\circ }\right)$

  $2\left(\cos {174}^{\circ }+i\sin {174}^{\circ }\right)$

  $2\left(\cos {246}^{\circ }+i\sin {246}^{\circ }\right)$

  $2\left(\cos {318}^{\circ }+i\sin {318}^{\circ }\right)$

Graph:

  

Interpretation:

The graph of the fifth roots

  $2\left(\cos {30}^{\circ }+i\sin {30}^{\circ }\right),2\left(\cos {102}^{\circ }+i\sin {102}^{\circ }\right),2\left(\cos {174}^{\circ }+i\sin {174}^{\circ }\right),2\left(\cos {246}^{\circ }+i\sin {246}^{\circ }\right)$ and $2\left(\cos {318}^{\circ }+i\sin {318}^{\circ }\right)$ are sketched as above.