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Math Help - prove |cos x| <= |x|

  1. #1
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    Exclamation prove |cos x| <= |x|

    Please give me the proof for " absolute value of cos x is less than or equal to absolute value of x? "
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    Re: prove |cos x| <= |x|

    Quote Originally Posted by maddymath View Post
    Please give me the proof for " absolute value of cos x is less than or equal to absolute value of x? "
    You can't. Take for example x = 0...
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  3. #3
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    Re: prove |cos x| <= |x|

    Have you ever looked at graphs of y= cos(x) and y= |x|?
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    Re: prove |cos x| <= |x|

    maybe ...

    |\sin{x}| \le |x|

    ... ???
    Attached Thumbnails Attached Thumbnails prove |cos x| &lt;= |x|-absgraph.jpg  
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    Re: prove |cos x| <= |x|

    thanks guys you gave me confidence actually it were written in a professor's notes but now I'm sure |cos x| =! |x| but |sin x| = |x|
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    Re: prove |cos x| <= |x|

    Quote Originally Posted by maddymath View Post
    I'm sure |cos x| =! |x| but |sin x| = |x|
    Hmm, look at skeeter's graph again...
    Last edited by emakarov; July 6th 2012 at 01:17 AM.
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    Re: prove |cos x| <= |x|

    Quote Originally Posted by maddymath View Post
    thanks guys you gave me confidence actually it were written in a professor's notes but now I'm sure |cos x| =! |x| but |sin x| = |x|
    Last time I checked, \displaystyle \begin{align*} \sin{\frac{\pi}{4}} \neq \frac{\pi}{4} \end{align*}, for example...
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    Re: prove |cos x| <= |x|

    Quote Originally Posted by maddymath View Post
    thanks guys you gave me confidence actually it were written in a professor's notes but now I'm sure |cos x| =! |x| but |sin x| = |x|
    I assume you meant "|sin(x)|< |x|", not equal.
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    Re: prove |cos x| <= |x|

    sin(x) is approximately x, when |x| is small enough. this is because the 2 curves g(x) = sin(x) and f(x) = x, have "the same slope and value" at the point x = 0.
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    Re: prove |cos x| <= |x|

    sin pi/4 =! pi/4
    but
    sin pi/4 < pi/4
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  11. #11
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    Re: prove |cos x| <= |x|

    Here is a small problem which is related to the given question. prove y=f(x)=x-sinx >=0 for all x>=0

    take dy/dx=1-cosx >=0 for all x, y is increasing func, f(x) >= f(0) = 0 for all x>=0

    then we can easily extend this solution to solve |sinx| <= |x|
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    Re: prove |cos x| <= |x|

    Quote Originally Posted by panda89 View Post
    Here is a small problem which is related to the given question. prove y=f(x)=x-sinx >=0 for all x>=0

    take dy/dx=1-cosx >=0 for all x, y is increasing func, f(x) >= f(0) = 0 for all x>=0

    then we can easily extend this solution to solve |sinx| <= |x|
    I got what I wanted guys thanks
    but isn't this amazing see graph in this link graph of sin - Google Search

    in this graph sinx does never equal 0,
    except only at zero
    I was expecting it to be zero at pi, 2pi, 3pi ..... and so on, everywhere on (i-e n(pi)) where n belongs to Z+
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  13. #13
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    Re: prove |cos x| <= |x|

    Quote Originally Posted by maddymath View Post
    but isn't this amazing see graph in this link graph of sin - Google Search

    in this graph sinx does never equal 0,
    except only at zero
    I don't see what you are describing in the images that this Google search brings up.
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  14. #14
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    Re: prove |cos x| <= |x|

    Quote Originally Posted by emakarov View Post
    I don't see what you are describing in the images that this Google search brings up.
    its a graph of sinx we can move our mouse cursor to change the value of x
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    Re: prove |cos x| <= |x|

    Quote Originally Posted by maddymath View Post
    its a graph of sinx we can move our mouse cursor to change the value of x
    But I don't see that "this graph sinx does never equal 0, except only at zero," as you write in post #12. Perhaps you need to change the zoom.
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