Question

I'm not sure I understand how to do this. Thanks for the help!

Find all seventh roots of unity and sketch them on the aces below.

http://www.google.com/imgres?imgurl=http%3A%2F%2Fuser-content.enotes.com%2F7c4540d5f581a45d753fc2cff4180d9b3cb69630_thumb.png&imgrefurl=http%3A%2F%2Fwww.enotes.com%2Fhomework-help%2Ffind-all-seventh-roots-unity-sketch-them-axes-436533&h=287&w=282&tbnid=aGzfxmKJbQW5JM%3A&zoom=1&docid=WUxe1gxLnyuV-M&ei=pg55U7atMtOSqAbAgYGIAQ&tbm=isch&client=safari&ved=0CFUQMygBMAE&iact=rc&uact=3&dur=849&page=1&start=0&ndsp=32

Answer 1

axes not aces

Answer 2

@nincompoop @Michele_Laino @Nnesha @Luigi0210 @whpalmer4 @uri @jhonyy9 @sleepyjess

Answer 3

Man I know how to do this, but not sure how to explain it /:

Answer 4

We want to solve the equation \( \large z^7 = 1 \)
$$\large{
1= \cos(2\pi k) + i \sin(2\pi k ), k \in \mathbb Z
\\\text {by substitution}
\\z^7= \cos(0+2\pi k) + i \sin(0+2\pi k )
\\\text {by demoivres theorem}
\\z^7= \cos(2\pi k/7) + i \sin(2\pi k/7 ), k = 0,1,2,3,4,5,6
}$$

Answer 5

Once I have all that, it should look like this: https://www.wolframalpha.com/input/?i=x^7-1%3D0 ?

Answer 6

For the bottom graph

Answer 7

If you click on approximate form you can get the decimal values.

Answer 8

Is it okay if I have help with the graphing?

Answer 9

yes

Answer 10

there is a picture here
http://prntscr.com/7h01sn

Answer 11

But it doesn't have the radian values. Is there a way to graph it with the radians there?

Answer 12

You mean on your graph? There isn't shown pi/7.

Answer 13

The best you can do is estimate where to plot it

Answer 14

Okay

Answer 15

the first point is (1,0) that is clear

Answer 16

then the next point is (cos 2pi/7, sin(2pi/7))
where is the angle 2pi/7 relative to pi/6 and pi/6

Answer 17

all the points will be on the unit circle

Answer 18

Would I draw it at the first mini circle in this graph?

Answer 19

The dot is (1,0)?

Answer 20

http://prntscr.com/7h04hm

Answer 21

okay, thanks

Answer 22

The rest are drawn around the 1 area?

Answer 23

thats correct.
It might be easier to conver the radian angles to degrees
2pi/7 * 180/pi = 51.43 degrees approximately

Answer 24

Should I draw a line connecting the dot?

Answer 25

pi/4 is 45 degrees
pi/3 is 60 degrees
So the second point will be between these two angles.

Answer 26

52.5 ok

Answer 27

I believe we should just plot the points. But if you do connect the points you will get a 7 sided regular polygon, a heptagon.

Answer 28

I meant to the center, like wolfram. Do you think that would be fine, or should I leave it be?

Answer 29

http://prntscr.com/7h06t4

Answer 30

I don't see wolfram plotting to the center, can you show me?

Answer 31

There is a line connecting the dots to the center. It's grey.

Answer 32

I see it now. Those are optional.

Answer 33

The main thing are the points.

Answer 34

okay, cool

Answer 35

Can you help me with the rest of the points?

Answer 36

convert the radian angles to degrees, that might be easier

Answer 37

Cool. Thanks for the help! I gave you a medal :)

Answer 38

Your welcome :)