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Inequality (Posted on 2015-02-11) Difficulty: 2 of 5
Show that (sin x)^(sin x) < (cos x)^(cos x) when 0 < x < pi/4

No Solution Yet Submitted by Danish Ahmed Khan    
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Trigonometric solution | Comment 2 of 5 |
Well, of course it is.
Because in that range, sin(x) < cos(x).

Consider a right triangle with acute angles x and y, and with the sides that are opposite them having length X and Y respectively.
As long as x < y, then X < Y.  Call the length of the hypoteneuse Z.  Y is the side that is adjacent to x, so sin(x) = X/Z and cos(x) = Y/Z.  Therefore, if x is between 0 and 45 degrees, sin(x) < cos(x).

q.e.d. 


  Posted by Steve Herman on 2015-02-11 17:54:20
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