# Thread: prove |cos x| <= |x|

1. ## 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? "

2. ## Re: prove |cos x| <= |x|

Originally Posted by maddymath
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...

3. ## Re: prove |cos x| <= |x|

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

4. ## Re: prove |cos x| <= |x|

maybe ...

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

... ???

5. ## 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|

6. ## Re: prove |cos x| <= |x|

Originally Posted by maddymath
I'm sure |cos x| =! |x| but |sin x| = |x|
Hmm, look at skeeter's graph again...

7. ## Re: prove |cos x| <= |x|

Originally Posted by maddymath
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...

8. ## Re: prove |cos x| <= |x|

Originally Posted by maddymath
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.

9. ## 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.

10. ## Re: prove |cos x| <= |x|

sin pi/4 =! pi/4
but
sin pi/4 < pi/4

11. ## 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|

12. ## Re: prove |cos x| <= |x|

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+

13. ## Re: prove |cos x| <= |x|

Originally Posted by maddymath
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.

14. ## Re: prove |cos x| <= |x|

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

15. ## Re: prove |cos x| <= |x|

Originally Posted by maddymath
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.