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|Trigo inequality (Posted on 2014-12-11)
Let c be the Champernowne's constant,
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 – x3/6,
cos(x) > 1 – x2/2,
tan(x) > x + x3/3
So sin(x) + cos(x) + tan(x) > 1 +
2x - x2/2 + x3/6
> 1 + 2x – x2/2
& we need to prove that f(x) = 1 + 2x – x2/2 - 10x > 0 when x
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 2014-12-14 11:31:49
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