## anonymous one year ago 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

1. anonymous

axes not aces

2. anonymous

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

3. anonymous

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

4. anonymous

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

5. anonymous

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

6. anonymous

For the bottom graph

7. anonymous

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

8. anonymous

Is it okay if I have help with the graphing?

9. anonymous

yes

10. anonymous

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

11. anonymous

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

12. anonymous

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

13. anonymous

The best you can do is estimate where to plot it

14. anonymous

Okay

15. anonymous

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

16. anonymous

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

17. anonymous

all the points will be on the unit circle

18. anonymous

|dw:1434318947436:dw| Would I draw it at the first mini circle in this graph?

19. anonymous

|dw:1434319023208:dw|

20. anonymous

The dot is (1,0)?

21. anonymous
22. anonymous

okay, thanks

23. anonymous

The rest are drawn around the 1 area?

24. anonymous

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

25. anonymous

Should I draw a line connecting the dot?

26. anonymous

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

27. anonymous

52.5 ok

28. anonymous

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.

29. anonymous

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

30. anonymous
31. anonymous

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

32. anonymous

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

33. anonymous

I see it now. Those are optional.

34. anonymous

The main thing are the points.

35. anonymous

okay, cool

36. anonymous

Can you help me with the rest of the points?

37. anonymous

convert the radian angles to degrees, that might be easier

38. anonymous

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

39. anonymous