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Trigo inequality (Posted on 20141211) 

Let c be the Champernowne's constant,
or c=0.123456789101112131415161718192021....
Show, without calculator aid, that
sin(c) + cos(c) + tan(c) > 10c
including cubics  spoiler

 Comment 10 of 11 

I agree with Charlie..
All the terms in the
expansion of tan(x) are positive, so including
its cubed term should solve the problem.
sin(x) > x – x^{3}/6,
cos(x) > 1 – x^{2}/2,
tan(x) > x + x^{3}/3
So sin(x) + cos(x) + tan(x) > 1 +
2x  x^{2}/2 + x^{3}/6
> 1 + 2x – x^{2}/2
& we need to prove that f(x) = 1 + 2x – x^{2}/2  10x > 0 when x
= c.
f(x) is a positive decreasing function of x between 0 and its
positive root, and f(0.124) = 0.000312 > 0 (w/o calculator).
Since c < 0.124, we can be sure that f(c) > 0.

Posted by Harry
on 20141214 11:31:49 


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