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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 No Subject | Comment 5 of 11 |

First, note that since sin(x) is between -1 and 1, and cos(y) is positive for y between -1 and 1, so cos(sin(x)) is positive for all x.

Next, consider the following:

D = {cos (sin x)}2 - {sin (cos x)}2

= 1 - {sin (sin x)}2 - {sin (cos x)}2.

Note that for all x, (sin x)2 <= [less than or equal to] x2, so

D >= 1 - (sin x)2 - (cos x)2 = 0.

Thus, since cos(sin x) > 0, this means

cos(sin x) >= sin(cos x) for all x.

Also, it follows that (sin x)2 = x2 only for x = 0, and

cos(sin 0) = 1 > sin(cos 0) ~ .8415

Therefore, cos(sin x) > sin(cos x) for all real x.


  Posted by Josh70679 on 2005-08-12 23:06:44
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