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Math Help - Trigonometric inequality bounded by lines

  1. #1
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    Trigonometric inequality bounded by lines

    How can it be shown that 16x\cos(8x)+4x\sin(8x)-2\sin(8x)<|17x|?


    This problem arises from work with damped motion in spring-mass systems in Differential Equations. I have gotten to this inequality after some algebraic manipulation, but am completely stuck here.


    Here is the illustrative graph provided by Wolfram Alpha:

    http://i.stack.imgur.com/oWf9E.png


    Thanks!
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  2. #2
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    Re: Trigonometric inequality bounded by lines

    Quote Originally Posted by abscissa View Post
    How can it be shown that 16x\cos(8x)+4x\sin(8x)-2\sin(8x)<|17x|?


    This problem arises from work with damped motion in spring-mass systems in Differential Equations. I have gotten to this inequality after some algebraic manipulation, but am completely stuck here.


    Here is the illustrative graph provided by Wolfram Alpha:

    http://i.stack.imgur.com/oWf9E.png


    Thanks!
    You looking at the envelope of a sinusoidal waveform. So treat cos(8x) and sin(8x) as basis functions and find the envelope by taking the magnitude of the overall expression.

    You have

    a\cos(8x)+b\sin(8x)$ where $a=16x$ and $b=4x-2

    The envelope is given by \sqrt{a^2+b^2}


    Trigonometric inequality bounded by lines-clipboard01.jpg

    You wish to show this envelope satisfies your original inequality. See if you can work it now.
    Thanks from abscissa
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  3. #3
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    Re: Trigonometric inequality bounded by lines

    ROMSEK

    Check your calculations . A blow up of the graph indicates that this inequality is not valid for x=-238 to-0.337...

    see below
    Trigonometric inequality bounded by lines-trigonometry-graph.png
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  4. #4
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    Re: Trigonometric inequality bounded by lines

    Quote Originally Posted by MINOANMAN View Post
    ROMSEK

    Check your calculations . A blow up of the graph indicates that this inequality is not valid for x=-238 to-0.337...

    see below
    Click image for larger version. 

Name:	TRIGONOMETRY GRAPH.PNG 
Views:	10 
Size:	56.5 KB 
ID:	29995
    Your graph seems to intersect (0,0). It shouldn't. The y intercept of the envelope is at 2. I think mine is ok.
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  5. #5
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    Re: Trigonometric inequality bounded by lines

    Maybe what Minoanman meant was that the envelope is not always less than or equal to |17x|. Here's the graph of |17x|-envelope:

    Trigonometric inequality bounded by lines-mhftrig3.png

    So I think there's a little more work to do for the original inequality.
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  6. #6
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    Re: Trigonometric inequality bounded by lines

    Quote Originally Posted by johng View Post
    Maybe what Minoanman meant was that the envelope is not always less than or equal to |17x|. Here's the graph of |17x|-envelope:

    Click image for larger version. 

Name:	MHFtrig3.png 
Views:	7 
Size:	14.4 KB 
ID:	29996

    So I think there's a little more work to do for the original inequality.
    I see. Looking at the graph of |17x| - the raw signal we see that the inequality isn't valid near -0.3 in which case the envelope won't satisfy the inequality either. That means the problem assertion is incorrect.

    Trigonometric inequality bounded by lines-clipboard01.jpg
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  7. #7
    Senior Member sakonpure6's Avatar
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    Re: Trigonometric inequality bounded by lines

    Out of curiosity, what level of education is this at (University? what course?)
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