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Math Help - Proof

  1. #1
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    Proof

    I think this goes in this section.

    Let x be a real number. Prove that

    abs(sin nx) <= n*abs(sinx) for all positive integers n.
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  2. #2
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    Quote Originally Posted by Nichelle14
    I think this goes in this section.

    Let x be a real number. Prove that

    abs(sin nx) <= n*abs(sinx) for all positive integers n.
    I did not try to do the problem yet but you can use the fact that,
    \sqrt{x^2}=|x|
    Thus, you need to prove that,
    \sqrt{\sin ^2 nx}\leq n\sqrt{\sin^2 x}
    If and only if,
    \sin^2 nx\leq n^2\sin ^2 x
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  3. #3
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    I still don't understand what to do.
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  4. #4
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    Mathematical induction is the tool here.
    It is true for n=1 thus, there is a k such as, |\sin kx|\leq k|\sin x|
    Thus,
    0\leq |\sin kx|\leq k|\sin x|
    But,
    0\leq |\cos x|\leq 1
    Thus, (mutiply inequalities notice they are non-negative),
    |\sin kx||\cos x|\leq k|\sin x|......(1)
    Also,
    0\leq |\sin x|\leq |\sin x|
    And,
    0\leq |\cos kx|\leq 1
    Thus, (mutiply inequalites notice they are non-negative),
    |\cos kx||\sin x|\leq |\sin x|......(2)
    Using the property |x||y|=|xy| on (1) and (2) we have,
    |\sin kx \cos x|\leq k|\sin x|
    |\cos kx \sin x|\leq |\sin x|
    Now, add these inequalities,
    |\sin kx \cos x|+|\cos kx \sin x|\leq k|\sin x|+|\sin x|
    Now, by triangular inequality,
    |\sin kx \cos x|+|\cos kx \sin x|\geq |\sin kx \cos x+\cos kx \sin x|
    By transitivity ( a<b \mbox{ and }b<c \rightarrow a<c),
    |\sin kx \cos x+\cos kx \sin x|\leq k|\sin x|+|\sin x|
    Thus, recognizing the sum for sine and simply the right side,
    |\sin (k+1) x|\leq (k+1)|\sin x|
    Proof is complete.
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  5. #5
    MHF Contributor Quick's Avatar
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    Is it possible to say 5\leq5
    (because it seems somewhat ridiculous)
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  6. #6
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    Quote Originally Posted by Quick
    Is it possible to say 5\leq5
    (because it seems somewhat ridiculous)
    Of course.

    a\leq b is defined to be true whenever,
    a<b OR a=b.
    Furthermore, in mathematics this expression is used a lot. There is one powerful theorem in set theory (Zorn's Lemma) which is based on the fact that a\leq a .
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