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Math Help - Proof

  1. #1
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    Proof

    Let t be a fixed number and let c=cos(t) and s=sin(t) and \mathbf{x}=(c, cs, cs^2,..., cs^{n-1}, s^n)^T.

    Show that \mathbf{x} is a unit vector.

    Hint: 1+s^2+s^4+...+s^{2n-2}=\frac{1-s^{2n}}{1-s^2}

    Not sure how that even helps unless I want to the hint via induction.
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  2. #2
    Super Member Failure's Avatar
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    Quote Originally Posted by dwsmith View Post
    Let t be a fixed number and let c=cos(t) and s=sin(t) and \mathbf{x}=(c, cs, cs^2,..., cs^{n-1}, s^n)^T.

    Show that \mathbf{x} is a unit vector.

    Hint: 1+s^2+s^4+...+s^{2n-2}=\frac{1-s^{2n}}{1-s^2}

    Not sure how that even helps unless I want to the hint via induction.
    Yes, sure it helps, for the square of the length of that vector is
    c^2+c^2s^2+\cdots+c^2s^{2n-2}+s^{2n}=c^2\frac{1-s^{2n}}{1-s^2}+s^{2n}=1-s^{2n}+s^{2n}=1
    since 1-s^2=c^2.
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  3. #3
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    Quote Originally Posted by Failure View Post
    Yes, sure it helps, for the square of the length of that vector is
    c^2+c^2s^2+\cdots+c^2s^{2n-2}+s^{2n}=c^2\frac{1-s^{2n}}{1-s^2}+s^{2n}=1-s^{2n}+s^{2n}=1
    since 1-s^2=c^2.
    How did you get c^2?
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  4. #4
    Super Member Failure's Avatar
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    Quote Originally Posted by dwsmith View Post
    How did you get c^2?
    Sorry, but I don't quite understand your question.
    I just wrote down what, imho, the square of the length of that vector (c,cs,cs^2,\ldots,cs^{n-1},s^n) happens do be: namely the sum of the squares of its coordinates. Hence I get
    c^2+(cs)^2+\cdots+(cs^{n-1})^2+(s^n)^2=c^2+c^2s^2+\cdots+c^2s^{2n-2}+s^{2n}
    Can you be a bit more specific as to where, exactly the occurrence of c^2 in what I wrote surprises you?
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  5. #5
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    If you are referring to his " 1- s^2= c^2, remember that you said "let c= cos(t) and s= sin(t)". Failure (he really should change that user name!) is just using the fact that sin^2(t)+ cos^2(t)= 1.
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