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**RABIT****Guest**

**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

I think the folllowing formula might prove handy:

It works for all θ and can be easily proved.

If it doesnt work, try something else.

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**Stanley_Marsh****Member**- Registered: 2006-12-13
- Posts: 345

I know a geometry method , you can draw a regular heptagon , than each exterior angle is

Let each side be 1 , and let a vertex be origin , then vector

I don't know what to do after that . I remember there is a method like this.

Numbers are the essence of the Universe

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**JaneFairfax****Member**- Registered: 2007-02-23
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Wait! I just thought of another method, using complex numbers.

Consider the roots of the complex equation

The roots are

The sum of the roots must be 0. (In general, the sum of the roots of a polynomial equation of degree *n* is given by −(coefficient of *z*[sup]*n*−1[/sup]) divided by coefficient of *z*[sup]*n*[/sup].)

If you sum the roots, the imaginary parts of the complex conjugates cancel in pairs, leaving twice the real part, i.e.

Note that

And thats the problem solved!

*Last edited by JaneFairfax (2007-03-21 09:57:17)*

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**Stanley_Marsh****Member**- Registered: 2006-12-13
- Posts: 345

How do I upload picture , I pressed the upload botton , and select my picture but it didnt show.

Numbers are the essence of the Universe

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**mathsyperson****Moderator**- Registered: 2005-06-22
- Posts: 4,900

It looks like you've tried to upload an image using the [img] tags. Those are used to show images that are already on the internet, by typing the URL of the image between them.

To upload an image from your computer, you use the image upload box that should appear below the box you type in when posting. Change the number in there from 0 to however many images you want to show, then it should be a simple case of browsing your computer and selecting each one you want to upload. If that's what you were doing, maybe the image didn't fit the restrictions?

Why did the vector cross the road?

It wanted to be normal.

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**Stanley_Marsh****Member**- Registered: 2006-12-13
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Janefairfax's proof is fantastic . I think it can also be solved by geometry , This maybe will give you a hint , ACO is part of a Heptagon , and BA is the extend of AC , CO vertical to BO , and Ao is one side of the Heptagon , length is 1. I didnt actually prove it , but Can this work?

Numbers are the essence of the Universe

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

The proof using complex numbers may well have a geometric interpretation, so using geometry to prove the result is probably equivalent (and probably easier to understand for people not so familiar with complex numbers).

PS: Yes, if you put a regular heptagon on the complex plane with its centre at the origin and one vertex at −1, the vertices of the heptagon will correspond to the roots of the complex equation *z*[sup]7[/sup] + 1 = 0

*Last edited by JaneFairfax (2007-03-22 02:09:06)*

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**RABIT****Guest**

who can give more ? i am in grade 11 and i dont know complex number and these stuff

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