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Math Help - Proof of regular polyon inscribed in a cricle

  1. #1
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    Proof of regular polyon inscribed in a cricle

    Hello ,


    I wanted to find a relationship between the number of sides of a regular polygon inscribed in a unit circle and the length of side of the polygon (The vertices are roots of equation in the form of Z^n=1).


    I came up with two equation that Length = 2sin(pi/n) and Length= Sqrt(2-2cos(2pi/n) , where n is the number of sides.



    I was able to prove the two roles geometrically ,


    But I need to prove them , or at least one of them analytically (e.g. using algebra)


    Can you help me?
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    Re: Proof of regular polyon inscribed in a cricle

    Quote Originally Posted by Hyunqul View Post
    Hello ,


    I wanted to find a relationship between the number of sides of a regular polygon inscribed in a unit circle and the length of side of the polygon (The vertices are roots of equation in the form of Z^n=1).


    I came up with two equation that Length = 2sin(pi/n) and Length= Sqrt(2-2cos(2pi/n) , where n is the number of sides.



    I was able to prove the two roles geometrically ,


    But I need to prove them , or at least one of them analytically (e.g. using algebra)


    Can you help me?
    I really don't have the patience to write it all out again. Read Lesson 7 in this thread...

    Harry Potter Forums :: The Slug Club - Arithmancy Master Class (Private) :: HP Community & Forums
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    Re: Proof of regular polyon inscribed in a cricle

    Pardon ?
    May I repeat my questions ... I'm looking for porving the relationship bewteen the number of sides and length of one side of a regular polygon inscribed inside a unit circle of radius (1). The rules I came with were : Length = 2sin(pi/n) and Length= Sqrt(2-2cos(2pi/n). I've made the gemoetric proofs....and looking for alegbriac proofs....Do they exist in the link you offered me ? Can you instruct me to their location ?
    thanks.
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    Re: Proof of regular polyon inscribed in a cricle

    Quote Originally Posted by Hyunqul View Post
    Pardon ?
    May I repeat my questions ... I'm looking for porving the relationship bewteen the number of sides and length of one side of a regular polygon inscribed inside a unit circle of radius (1). The rules I came with were : Length = 2sin(pi/n) and Length= Sqrt(2-2cos(2pi/n). I've made the gemoetric proofs....and looking for alegbriac proofs....Do they exist in the link you offered me ? Can you instruct me to their location ?
    thanks.
    Look for Lesson 7, it's on that page... If you understand what is being done in that lesson, then a single extra step will complete your proof... It will not involve sines or cosines.
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    Re: Proof of regular polyon inscribed in a cricle

    Quote Originally Posted by Hyunqul View Post
    length of one side of a regular polygon inscribed inside a unit circle of radius (1). The rules I came with were : Length = 2sin(pi/n) and Length= Sqrt(2-2cos(2pi/n).
    Have a look at this webpage.
    Look at equation (6).
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    Re: Proof of regular polyon inscribed in a cricle

    Hmmmm..I'm sorry Gentelmen , I cant quiet get what you presented.

    If you could provide more calrification of ALGEBRAIC PROOF ... That would be vey generous of you ...otherwise, thanks anyway.
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    Re: Proof of regular polyon inscribed in a cricle

    \
    Quote Originally Posted by Hyunqul View Post
    If you could provide more calrification of ALGEBRAIC PROOF ... That would be vey generous of you ...otherwise, thanks anyway.
    In the diagram I have added a perpendicular bisector of the cord a.
    That happens to also bisect the angle \theta.
    In either of the two smaller triangles \sin\left(\frac{\theta}{2}\right)=\frac{a}{2R}.
    That gives us a=2R\sin\left(\frac{\theta}{2}\right).
    Of course here R=1~\&~\theta\text=\left(\frac{2\pi}{n}\right).
    Now you have the length of a side.
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    Re: Proof of regular polyon inscribed in a cricle

    hmmmm...thanks sir , but my aim was an algebraics proof....a mostly analytical one (e.g.using trig identities).
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