Chapter 6.6, Problem 74E

Solution

a)

To visualize: The eight-eight roots of the unity.

The traced points are shown in figure (1).

Given information:

It is given that $x=\cos \left(\left(\frac{2\pi }{8}\right)t\right)$ and $y=\sin \left(\left(\frac{2\pi }{8}\right)t\right)$ .

Calculation:

By following the instructions, calculator should display $8$ connected points representing the $8$ eighth roots of unity.

Using the TRACE feature gives the roots where $\left(x,y\right)$ represents the complex number $x+iy$ .

Trace the points $1,0.71+0.71i,i,-0.71+0.71i,-1,-0.71-0.71i,-i$ and $0.71-0.71i$ .

  

Figure (1)

Therefore, the traced points are shown in figure (1).

b)

To find: Whether the other generates the eighth roots of unity.

Yes, the other generates the eighth roots of unity.

Calculation:

The arguments of the eighth roots differ by $\frac{2\pi }{8}$ so the following $7$ arguments.

  $\frac{4\pi }{8},\frac{6\pi }{8},\frac{8\pi }{8},\frac{10\pi }{8},\frac{12\pi }{8},\frac{14\pi }{8},\text{and},\frac{16\pi }{8}$

For $\frac{4\pi }{8}$ ,

Enter the number as shown below.

  

Press the GRAPH button to obtain the graph as shown below.

  

For $\frac{6\pi }{8}$ ,

Enter the number as shown below.

  

Press the GRAPH button to obtain the graph as shown below.

  

For $\frac{8\pi }{8}$ ,

Enter the number as shown below.

  

Press the GRAPH button to obtain the graph as shown below.

  

For $\frac{10\pi }{8}$ ,

Enter the number as shown below.

  

Press the GRAPH button to obtain the graph as shown below.

  

For $\frac{12\pi }{8}$ ,

Enter the number as shown below.

  

Press the GRAPH button to obtain the graph as shown below.

  

For $\frac{14\pi }{8}$ ,

Enter the number as shown below.

  

Press the GRAPH button to obtain the graph as shown below.

  

For $\frac{16\pi }{8}$ ,

Enter the number as shown below.

  

Press the GRAPH button to obtain the graph as shown below.

  

Therefore, yes, the other generates the eighth roots of unity.

c)

To repeat: The above process for the fifth, sixth, and seventh roots of unity.

All the roots are traced below.

Given information:

The complex number is ${\left(1+i\sqrt{3}\right)}^3$ .

Calculation:

For the fifth roots, use the argument $\frac{2\pi }{5}$ . The arguments of the fifth roots differ by $\frac{2\pi }{5}$ so use the following $4$ arguments:

  $\frac{4\pi }{5},\frac{6\pi }{5},\frac{8\pi }{5},\frac{10\pi }{5}$

Among these, three more arguments generate the fifth roots of unity which are as follows.

  $\frac{4\pi }{5},\frac{6\pi }{5},\frac{8\pi }{5}$

For $\frac{2\pi }{5}$ ,

Enter the number as shown below.

  

Press the GRAPH button to obtain the graph as shown below.

  

For $\frac{4\pi }{5}$ ,

Enter the number as shown below.

  

Press the GRAPH button to obtain the graph as shown below.

  

For $\frac{6\pi }{5}$ ,

Enter the number as shown below.

  

Press the GRAPH button to obtain the graph as shown below.

  

For $\frac{8\pi }{5}$ ,

Enter the number as shown below.

  

Press the GRAPH button to obtain the graph as shown below.

  

For $\frac{10\pi }{5}$ ,

Enter the number as shown below.

  

Press the GRAPH button to obtain the graph as shown below.

  

For the sixth roots, use the argument $\frac{2\pi }{6}$ . The arguments of the sixth roots differ by $\frac{2\pi }{6}$ so use the following 5 arguments.

  $\frac{4\pi }{6},\frac{6\pi }{6},\frac{8\pi }{6},\frac{10\pi }{6},\frac{12\pi }{6}$

Among these, one more arguments generate the sixth roots of unity which is as follows.

  $\frac{10\pi }{6}$

For $\frac{2\pi }{6}$ ,

Enter the number as shown below.

  

Press the GRAPH button to obtain the graph as shown below.

  

For $\frac{4\pi }{6}$ ,

Enter the number as shown below.

  

Press the GRAPH button to obtain the graph as shown below.

  

For $\frac{6\pi }{6}$ ,

Enter the number as shown below.

  

Press the GRAPH button to obtain the graph as shown below.

  

For $\frac{8\pi }{6}$ ,

Enter the number as shown below.

  

Press the GRAPH button to obtain the graph as shown below.

  

For $\frac{10\pi }{6}$ ,

Enter the number as shown below.

  

Press the GRAPH button to obtain the graph as shown below.

  

For $\frac{12\pi }{6}$ ,

Enter the number as shown below.

  

Press the GRAPH button to obtain the graph as shown below.

  

For the seventh roots, use the argument $\frac{2\pi }{7}$ . The arguments of the seventh roots differ by $\frac{2\pi }{7}$ so use the following 6 arguments.

  $\frac{4\pi }{7},\frac{6\pi }{7},\frac{8\pi }{7},\frac{10\pi }{7},\frac{12\pi }{7},\frac{14\pi }{7}$

Among these, five more arguments generate the seventh roots of unity which are as follows.

  $\frac{4\pi }{7},\frac{6\pi }{7},\frac{8\pi }{7},\frac{10\pi }{7},\frac{12\pi }{7}$

For $\frac{2\pi }{7}$ ,

Enter the number as shown below.

  

Press the GRAPH button to obtain the graph as shown below.

  

For $\frac{4\pi }{7}$ ,

Enter the number as shown below.

  

Press the GRAPH button to obtain the graph as shown below.

  

For $\frac{6\pi }{7}$ ,

Enter the number as shown below.

  

Press the GRAPH button to obtain the graph as shown below.

  

For $\frac{8\pi }{7}$ ,

Enter the number as shown below.

  

Press the GRAPH button to obtain the graph as shown below.

  

For $\frac{10\pi }{7}$ ,

Enter the number as shown below.

  

Press the GRAPH button to obtain the graph as shown below.

  

For $\frac{12\pi }{7}$ ,

Enter the number as shown below.

  

Press the GRAPH button to obtain the graph as shown below.

  

For $\frac{14\pi }{7}$ ,

Enter the number as shown below.

  

Press the GRAPH button to obtain the graph as shown below.

  

Therefore, all the roots are shown above.

d)

To write: The conjecture about an nth root of unity that generates all the nth roots of unity.

All the $n\text{th}$ roots of unity is either a unit circle or a rose curve with $n$ petals.

Calculation:

The $n\text{th}$ root of unity that generates all the $n\text{th}$ roots of unity is either a unit circle or a rose curve with $n$ petals, each having a length of $1$ . This makes sense since the modulus of unity is $1$ .

Therefore, all the $n\text{th}$ roots of unity is either a unit circle or a rose curve with $n$ petals.