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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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Solution Solution | Comment 8 of 10 |
 
Since the cos(x) and sin(x) are both periodic with a period
of 2 pi, we can restrict the domain to (-pi,pi].
Since cos(sin(x)) and sin(cos(x)) are both even, we can
further restrict the domain to [0,pi].
Since cos(sin(x)) >= 0 and sin(cos(x)) < 0 in (pi/2,pi],
we can further restrict our domain to [0,pi/2].
     f(x) = sin(x) + cos(x) >= 0
    f'(x) = cos(x) - sin(x) = 0 ==> x = pi/4
    f"(x) = -[sin(x) + cos(x)] < 0 at x = pi/4
     f(0) = f(pi/2) = 1
Therefore, in [0,pi/2]
  sin(x) + cos(x) <= sin(pi/4) + cos(pi/4) = sqrt(2) < pi/2
Since the sin(x) is strictly increasing in [0,pi/2],
  sin(cos(x)) < sin(pi/2 - sin(x)) = cos(sin(x))
Therefore, 
  cos(sin(x)) > sin(cos(x))
for all real x.
 
 

  Posted by Bractals on 2005-08-13 15:39:24
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