Trigonometric inequality bounded by lines

How can it be shown that ?

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

?

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

The envelope is given by

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.

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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?)