Chapter 6.6, Problem 116E

a.

To determine

To calculate:

The roots of complex number.

Answer to Problem 116E

The roots of the complex numbers are,

  $\begin{array}{l}\cos 22.5°+i\sin 22.5°\\ \cos 112.5°+i\sin 112.5°\\ \cos 202.5°+i\sin 202.5°\\ \cos 292.5°+i\sin 292.5°\end{array}$

Explanation of Solution

Given information:

Fourth roots of $i$ .

Formula used:

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

Where,

  $n=$ positive integer, $z=r\left(\cos \theta +i\sin \theta \right)$ is the complex number has exactly $n$ distinct $n^{th}$ roots, $k=0,1,\mathrm{2.....},n-1$

Calculation:

Consider the complex number $i$

Write in standard trigonometric form as

  $x^4=\cos {90}^{°}+i\sin 90°$

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

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

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

For getting the roots of the complex number using above formula,

  $\cos \frac{1}{4}\left(90°+360°k\right)+i\sin \frac{1}{4}\left(90°+360°k\right)$

So, for $k=0,1,2,3$ the roots are as follows:

  $\begin{array}{l}\cos 22.5°+i\sin 22.5°\\ \cos 112.5°+i\sin 112.5°\\ \cos 202.5°+i\sin 202.5°\\ \cos 292.5°+i\sin 292.5°\end{array}$

b.

To determine

To calculate:

Given roots in standard form.

Answer to Problem 116E

The complex roots are in the standard form as,

  $\begin{array}{l}\cos 22.5°+i\sin 22.5°=0.92+0.38i\\ \cos 112.5°+i\sin 112.5°=-0.38+0.92i\\ \cos 202.5°+i\sin 202.5°=-0.92-0.38i\\ \cos 292.5°+i\sin 292.5°=0.38-0.92i\end{array}$

Explanation of Solution

Given information:

Fourth roots of $i$ .

The roots are,

  $\begin{array}{l}\cos 22.5°+i\sin 22.5°\\ \cos 112.5°+i\sin 112.5°\\ \cos 202.5°+i\sin 202.5°\\ \cos 292.5°+i\sin 292.5°\end{array}$

Calculation:

Since, $\cos 22.5°=0.92$ and $\sin 22.5°=0.38$ write,

  $\cos 22.5°+i\sin 22.5°=0.92+0.38i$

Similarly, write

  $\begin{array}{l}\cos 112.5°+i\sin 112.5°=-0.38+0.92i\\ \cos 202.5°+i\sin 202.5°=-0.92-0.38i\\ \cos 292.5°+i\sin 292.5°=0.38-0.92i\end{array}$

c.

To determine

To graph:

The given roots graphically.

Explanation of Solution

Given information:

The roots are,

  $\begin{array}{l}\cos 22.5°+i\sin 22.5°\\ \cos 112.5°+i\sin 112.5°\\ \cos 202.5°+i\sin 202.5°\\ \cos 292.5°+i\sin 292.5°\end{array}$

Graph:

  

Interpretation:

In above graph, complex roots are described by the imaginary and real axis.