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Trigonometry nest (Posted on 2005-08-12) Difficulty: 4 of 5
Which is greater, sin(cos(x)) or cos(sin(x))? Prove it!

See The Solution Submitted by Federico Kereki    
Rating: 3.8333 (6 votes)

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Some Thoughts note the category; Close to a proof. | Comment 3 of 10 |
Since the category is "Calculus", no doubt there must be some way calculus can help prove the cos(sin(x)) is always greater than sin(cos(x)).

Let f(x) = cos(sin(x)) - sin(cos(x))
We suspect f(x) is always positive.  Let's find the minimum value of f(x).
Start by finding the first derivative, f'(x), and set it equal to zero.

f'(x) = -sin(sin(x)) * cos(x) + cos(cos(x)) * sin(x)
which I haven't been able to break down further.
But from graphing it out, f'(x) is equal to zero at a series of well defined values for x:
0, pi, 2pi, ..., k*pi    and
L, 2pi + or - L, 4pi + or - L, .....
where L is approx:  .6927.... something
(and probably at negative values of x too, but I didn't check that)

So, we now have a limited number of x values to check to proove that cos(sin) is always greater than sin(cos).

Not totally rigorous, partly uses numeric methods.  It's enough to convince me.

Hopefully someone can provide a complete pure mathematical proof.

  Posted by Larry on 2005-08-12 22:43:02
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