Chapter 14.5, Problem 15WE

To determine

To find: the required roots in $\text{x+iy}$ form.

Answer to Problem 15WE

  $\text{1+0i}$ , $\frac{\text{1}}{\text{2}}\text{+i}\frac{\sqrt{\text{3}}}{\text{2}}$ , $\text{-}\frac{\text{1}}{\text{2}}\text{+i}\frac{\sqrt{\text{3}}}{\text{2}}$ , $\text{-1+0i}$ , $\text{-}\frac{\text{1}}{\text{2}}\text{-i}\frac{\sqrt{\text{3}}}{\text{2}}$ and $\frac{\text{1}}{\text{2}}\text{-i}\frac{\sqrt{\text{3}}}{\text{2}}$ .

Explanation of Solution

Given:

The sixth root of unity.

Concept used:

If $\text{x}$ is the an $\text{nth}$ root of unity, then it satisfies the relation:

  ${\text{x}}^{\text{n}}\text{=1}$ .

  $\text{=cos0+isin0}$ .

  ${\text{x}}^{\text{n}}\text{=cos2kπ+isin2kπ}$ where $\text{k}$ is an integer.

Taking $\text{nth}$ root on both sides:

  ${\left({\text{x}}^{\text{n}}\right)}^{\frac{\text{1}}{\text{n}}}\text{=}{\left(\text{cos2kπ+isin2kπ}\right)}^{\frac{\text{1}}{\text{n}}}$ .

  $\text{x=}{\left(\text{cos2kπ+isin2kπ}\right)}^{\frac{\text{1}}{\text{n}}}$ .

According $\text{de moivre}$ s law $\text{x=}{\left(\text{cos2kπ+isin2kπ}\right)}^{\frac{\text{1}}{\text{n}}}$ can be written as:

In radian: $\text{x=}\left(\text{cos}\frac{\text{2kπ}}{\text{n}}\text{+isin}\frac{\text{2kπ}}{\text{n}}\right)$ .

In degree: $\text{x=}\left(\text{cos}\frac{\text{360k}}{\text{n}}\text{+isin}\frac{\text{360k}}{\text{n}}\right)$ .

Calculation:

The sixthroot of unity.

Using this formula:

  $\text{x=}\left(\text{cos}\frac{\text{360k}}{\text{n}}\text{+isin}\frac{\text{360k}}{\text{n}}\right)$ .

When $\text{k=0}$ ;

  $\text{x=}\left(\text{cos}\frac{\text{360×0}}{\text{6}}\text{+isin}\frac{\text{360×0}}{\text{6}}\right)\text{=1+0i}$ .

When $\text{k=1}$ :

  $\text{x=}\left(\text{cos}\frac{\text{360×1}}{\text{6}}\text{+isin}\frac{\text{360×1}}{\text{6}}\right){\text{=cos60}}^{\text{0}}{\text{+isin60}}^{\text{0}}\text{=}\frac{\text{1}}{\text{2}}\text{+i}\frac{\sqrt{\text{3}}}{\text{2}}$ .

When $\text{k=2}$ :

  $\text{x=}\left(\text{cos}\frac{\text{360×2}}{6}\text{+isin}\frac{\text{360×2}}{6}\right){\text{=cos120}}^{\text{0}}{\text{+isin120}}^{\text{0}}\text{=-}\frac{\text{1}}{\text{2}}\text{+i}\frac{\sqrt{\text{3}}}{\text{2}}$ .

When $\text{k=3}$

  $\text{x=}\left(\text{cos}\frac{\text{360×3}}{6}\text{+isin}\frac{\text{360×3}}{6}\right){\text{=cos180}}^{\text{0}}{\text{+isin180}}^{\text{0}}\text{=-1+0i}$ .

When $\text{k=4}$

  $\text{x=}\left(\text{cos}\frac{\text{360×4}}{\text{6}}\text{+isin}\frac{\text{360×4}}{\text{6}}\right){\text{=cos240}}^{\text{0}}{\text{+isin240}}^{\text{0}}\text{=-}\frac{\text{1}}{\text{2}}\text{-i}\frac{\sqrt{\text{3}}}{\text{2}}$ .

When $\text{k=5}$

  $\text{x=}\left(\text{cos}\frac{\text{360×5}}{\text{6}}\text{+isin}\frac{\text{360×5}}{\text{6}}\right){\text{=cos300}}^{\text{0}}{\text{+isin300}}^{\text{0}}\text{=}\frac{\text{1}}{\text{2}}\text{-i}\frac{\sqrt{\text{3}}}{\text{2}}$ .

Hence, the sixthroot of unity in $\text{x+iy}$ forms are:

  $\text{1+0i}$ , $\frac{\text{1}}{\text{2}}\text{+i}\frac{\sqrt{\text{3}}}{\text{2}}$ , $\text{-}\frac{\text{1}}{\text{2}}\text{+i}\frac{\sqrt{\text{3}}}{\text{2}}$ , $\text{-1+0i}$ , $\text{-}\frac{\text{1}}{\text{2}}\text{-i}\frac{\sqrt{\text{3}}}{\text{2}}$ and $\frac{\text{1}}{\text{2}}\text{-i}\frac{\sqrt{\text{3}}}{\text{2}}$ .