Chapter 6.5, Problem 140E

(a)

To determine

To find: The indicated root of the given expression

Answer to Problem 140E

The roots of the complex numbers are $2\left(\cos \frac{\pi }{6}+i\sin \frac{\pi }{6}\right)$ , $2\left(\cos \frac{17\pi }{30}+i\sin \frac{17\pi }{30}\right)$ , $2\left(\cos \frac{29\pi }{30}+i\sin \frac{29\pi }{30}\right)$ , $2\left(\cos \frac{41\pi }{30}+i\sin \frac{41\pi }{30}\right)$ , $2\left(\cos \frac{53\pi }{30}+i\sin \frac{53\pi }{30}\right)$ .

Explanation of Solution

Given information:

The given expression is:

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

Formula used:

Formula of $n^{th}$ root:

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

Consider the given expression as:

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

Consider the formula of $n^{th}$ root:

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

Substitute $\frac{5\pi }{6}$ for $\theta$ in the above expression:

  $\sqrt[5]{r}\left(\cos \frac{\frac{5\pi }{6}+2\pi k}{5}+i\sin \frac{\frac{5\pi }{6}+2\pi k}{5}\right)$

Substitute $0$ for $k$ in the above expression to find the first root:

  $\begin{array}{c}\sqrt[5]{r}\left(\cos \frac{\frac{5\pi }{6}+2\pi \left(0\right)}{5}+i\sin \frac{\frac{5\pi }{6}+2\pi \left(0\right)}{5}\right)\\ =2\left(\cos \frac{\pi }{6}+i\sin \frac{\pi }{6}\right)\end{array}$

Substitute $1$ for $k$ in the above expression to find the second root:

  $\begin{array}{c}\sqrt[5]{r}\left(\cos \frac{\frac{5\pi }{6}+2\pi \left(1\right)}{5}+i\sin \frac{\frac{5\pi }{6}+2\pi \left(1\right)}{5}\right)\\ =2\left(\cos \frac{17\pi }{30}+i\sin \frac{17\pi }{30}\right)\end{array}$

Substitute $2$ for $k$ in the above expression to find the third root:

  $\begin{array}{c}\sqrt[5]{r}\left(\cos \frac{\frac{5\pi }{6}+2\pi \left(2\right)}{5}+i\sin \frac{\frac{5\pi }{6}+2\pi \left(2\right)}{5}\right)\\ =2\left(\cos \frac{29\pi }{30}+i\sin \frac{29\pi }{30}\right)\end{array}$

Substitute $3$ for $k$ in the above expression to find the forth root:

  $\begin{array}{c}\sqrt[5]{r}\left(\cos \frac{\frac{5\pi }{6}+2\pi \left(3\right)}{5}+i\sin \frac{\frac{5\pi }{6}+2\pi \left(3\right)}{5}\right)\\ =2\left(\cos \frac{41\pi }{30}+i\sin \frac{41\pi }{30}\right)\end{array}$

Substitute $4$ for $k$ in the above expression to find the fifth root:

  $\begin{array}{c}\sqrt[5]{r}\left(\cos \frac{\frac{5\pi }{6}+2\pi \left(4\right)}{5}+i\sin \frac{\frac{5\pi }{6}+2\pi \left(4\right)}{5}\right)\\ =2\left(\cos \frac{53\pi }{30}+i\sin \frac{53\pi }{30}\right)\end{array}$

Hence, the roots of the complex numbers are $2\left(\cos \frac{\pi }{6}+i\sin \frac{\pi }{6}\right)$ , $2\left(\cos \frac{17\pi }{30}+i\sin \frac{17\pi }{30}\right)$ , $2\left(\cos \frac{29\pi }{30}+i\sin \frac{29\pi }{30}\right)$ , $2\left(\cos \frac{41\pi }{30}+i\sin \frac{41\pi }{30}\right)$ , $2\left(\cos \frac{53\pi }{30}+i\sin \frac{53\pi }{30}\right)$ .

(b)

To determine

To graph: The roots of the complex number from the above results.

Explanation of Solution

Given information:

The given expression is:

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

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

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

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

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

Graph:

Consider the given expression as:

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

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

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

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

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

Graph the above equations:

  

(c)

To determine

To find: The roots of the above expression in standard form

Answer to Problem 140E

The standard form of the complex numbers are $\sqrt{3}+i$ , $-0.4158+1.9563i$ , $-1.9890+0.2091i$ , $-0.8135-1.8271i$ and $1.4863-1.3383i$ .

Explanation of Solution

Given information:

The given expression is:

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

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

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

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

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

Consider the given expression as:

  $4\left(\cos 30°+i\sin 30°\right)$

  $4\left(\cos 210°+i\sin 210°\right)$

Substitute $\frac{\sqrt{3}}{2}$ for $\cos \frac{\pi }{6}$ and $\frac{1}{2}$ for $\cos \frac{\pi }{6}$ in the above expression:

  $\begin{array}{c}2\left(\cos \frac{\pi }{6}+i\sin \frac{\pi }{6}\right)=2\left(\frac{\sqrt{3}}{2}+\frac{1}{2}i\right)\\ =\sqrt{3}+i\end{array}$

Substitute $-0.2079$ for $\cos \frac{17\pi }{30}$ and $0.9781$ for $\sin \frac{17\pi }{30}$ in the above expression:

  $\begin{array}{c}2\left(\cos \frac{17\pi }{30}+i\sin \frac{17\pi }{30}\right)=2\left(-0.2079+0.9781i\right)\\ =-0.4158+1.9563i\end{array}$

Substitute $-0.9945$ for $\cos \frac{29\pi }{30}$ and $0.1045$ for $\sin \frac{29\pi }{30}$ in the above expression:

  $\begin{array}{c}2\left(\cos \frac{29\pi }{30}+i\sin \frac{29\pi }{30}\right)=2\left(-0.9945+0.1045i\right)\\ =-1.9890+0.2091i\end{array}$

Substitute $-0.4067$ for $\cos \frac{41\pi }{30}$ and $0.9135$ for $\sin \frac{41\pi }{30}$ in the above expression:

  $\begin{array}{c}2\left(\cos \frac{41\pi }{30}+i\sin \frac{41\pi }{30}\right)=2\left(-0.4067+0.9135i\right)\\ =-0.8135-1.8271i\end{array}$

Substitute $0.7431$ for $\cos \frac{53\pi }{30}$ and $-0.6691$ for $\sin \frac{53\pi }{30}$ in the above expression:

  $\begin{array}{c}2\left(\cos \frac{53\pi }{30}+i\sin \frac{53\pi }{30}\right)=2\left(0.7431+0.6691i\right)\\ =1.4863-1.3383i\end{array}$

Hence, the standard form of the complex numbers are $\sqrt{3}+i$ , $-0.4158+1.9563i$ , $-1.9890+0.2091i$ , $-0.8135-1.8271i$ and $1.4863-1.3383i$ .