Question

Complex number Question: Roots of Unity

Answer 1

By considering the seventh roots of unity, show that
\[\cos(\frac{ \pi }{ 7 })+\cos(\frac{ 3\pi }{ 7 })+ \cos(\frac{ 5\pi }{ 7 })=\frac{ 1 }{ 2 }\]

Answer 2

@Smexi_Girl Help me on this plz! I'm totally confused except I know the seven seventh roots of unity

Answer 3

aha finally a helper is online ! @terenzreignz

Answer 4

Ni hao comrade ^^
What seems to be the problem?

Answer 5

wait, you know Chinese?!

Answer 6

Not really, I'm just trying to get in the spirit of things, and anyway, I'm half-Chinese.

And don't change the subject XD

The seventh roots of unity?

Answer 7

Sorry hun, don't know this :(

Answer 8

well, i don't know how can the roots of unity be helpful

Answer 9

Me neither, but lay them on me anyway XD

Answer 10

My hair has roots of beauty if that helps XD

Answer 11

seventh roots of unit means all solutions to the following equation:
\[z ^{7}=1\]

Answer 12

Yup...

Answer 13

@Smexi_Girl Roots of beauty..... LOL!

Answer 14

so what ARE they?

Answer 15

:))

Answer 16

@caozeyuan focus.
The seventh roots of unit.

Answer 17

unity*

Answer 18

if my result is to be trusted, they should be:
\[z=e ^{\frac{ 2k \times \pi }{ 7 }*i}\]
where\[k \in \left[ 0,6 \right],k \in \mathbb{Z}^+0\]

Answer 19

Cannot see it. Do yourself a favour and type \Large before typing in equations XD
\[\Large z=e ^{\frac{ 2k \times \pi }{ 7 }*i}\]

Answer 20

Okay, this is better. But I prefer
\[\LARGE \text{cis}\left(\frac{2k\pi}{7}\right)\]

Answer 21

OK, same thing

Answer 22

back into question, how can i solve this sucky problem?

Answer 23

I'm thinking.

Answer 24

It might actually be better if I list them down, see if staring at them gives me insight:
\[\Large \begin{matrix}1\\\text{cis}\left(\frac{2\pi}{7}\right)\\\text{cis}\left(\frac{4\pi}{7}\right)\\\text{cis}\left(\frac{6\pi}{7}\right)\\\text{cis}\left(\frac{8\pi}{7}\right)\\\text{cis}\left(\frac{10\pi}{7}\right)\\\text{cis}\left(\frac{12\pi}{7}\right)\end{matrix}\]

Answer 25

actually I've been staring at them for 10 min before I put the question here.

Answer 26

Yes, well, let me stare at them too... LOL

Answer 27

Okay... let's just use this fact, maybe this helps:
\[\Large \text{cis}\left.(\theta\right.)=-\text{cis}(\theta -\pi)\]

Answer 28

Do I need to prove that?

Answer 29

no, obviously not

Answer 30

\[-\text{cis}\left(\frac{8\pi}{7}\right)-\text{cis}\left(\frac{10\pi}{7}\right)-\text{cis}\left(\frac{12\pi}{7}\right)=\text{cis}\left(\frac{\pi}{7}\right)+\text{cis}\left(\frac{3\pi}{7}\right)+\text{cis}\left(\frac{5\pi}{7}\right)\]

Answer 31

right, I can see a little clue here, but still dunno how to proceed

Answer 32

why are you " just looking around" ?

Answer 33

I'm still here. Don't nag me XD

Answer 34

let's continue then

Answer 35

I'm still thinking, don't rush me D:

Answer 36

Well, I'm going to sleep, it's 10 pm in BJ now

Answer 37

Hold on, I suggest you read this, it's rather concise.

Answer 38

http://math-kali.blogspot.com/2009/08/prove-cos-pi7-cos-3pi7-cos-5pi712.html

Answer 39

Good night ^^

Answer 40

I'm blocked, too bad!

Answer 41

I would be able to read it if I have a VPN