Let c be the Champernowne's constant, or c=0.123456789101112131415161718192021....

Show, without calculator aid, that
sin(c) + cos(c) + tan(c) > 10c

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 = 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 2014-12-14 11:31:49