# prove |cos x| <= |x|

• Jul 4th 2012, 01:15 AM
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? "
• Jul 4th 2012, 01:21 AM
Prove It
Re: prove |cos x| <= |x|
Quote:

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...
• Jul 4th 2012, 06:36 AM
HallsofIvy
Re: prove |cos x| <= |x|
Have you ever looked at graphs of y= cos(x) and y= |x|?
• Jul 4th 2012, 06:47 AM
skeeter
Re: prove |cos x| <= |x|
maybe ...

$\displaystyle |\sin{x}| \le |x|$

... ???
• Jul 5th 2012, 06:42 PM
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|
• Jul 6th 2012, 01:14 AM
emakarov
Re: prove |cos x| <= |x|
Quote:

I'm sure |cos x| =! |x| but |sin x| = |x|

Hmm, look at skeeter's graph again...
• Jul 6th 2012, 01:16 AM
Prove It
Re: prove |cos x| <= |x|
Quote:

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 \displaystyle \begin{align*} \sin{\frac{\pi}{4}} \neq \frac{\pi}{4} \end{align*}, for example...
• Jul 6th 2012, 05:36 AM
HallsofIvy
Re: prove |cos x| <= |x|
Quote:

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.
• Jul 6th 2012, 08:37 AM
Deveno
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.
• Jul 8th 2012, 08:57 AM
Re: prove |cos x| <= |x|
sin pi/4 =! pi/4
but
sin pi/4 < pi/4
• Jul 8th 2012, 10:20 AM
panda89
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|
• Jul 9th 2012, 10:53 AM
Re: prove |cos x| <= |x|
Quote:

Originally Posted by panda89
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+
• Jul 9th 2012, 11:04 AM
emakarov
Re: prove |cos x| <= |x|
Quote:

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.
• Jul 10th 2012, 08:36 AM
Re: prove |cos x| <= |x|
Quote:

Originally Posted by emakarov
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
• Jul 10th 2012, 08:40 AM
emakarov
Re: prove |cos x| <= |x|
Quote: