Trigonometric inequality bounded by lines

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

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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Re: Trigonometric inequality bounded by lines

Quote:

Originally Posted by

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

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

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

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

Attachment 29994

You wish to show this envelope satisfies your original inequality. See if you can work it now.

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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

Attachment 29995

Re: Trigonometric inequality bounded by lines

Quote:

Originally Posted by

**MINOANMAN** 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

Attachment 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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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:

Attachment 29996

So I think there's a little more work to do for the original inequality.

1 Attachment(s)

Re: Trigonometric inequality bounded by lines

Quote:

Originally Posted by

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

Attachment 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.

Attachment 29998

Re: Trigonometric inequality bounded by lines

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