Let c be the Champernowne's constant,
or c=0.123456789101112131415161718192021....
Show, without calculator aid, that
sin(c) + cos(c) + tan(c) > 10c
sin(c)>c-(c^3)/6
cos(c)>1-.5*c^2
tan(c)>sin(c)> c-(c^3)/6
The ">" sign results from ignoring the higher powers
in the Taylor's series
sin(c)+cos(c)+ tan(c)>1+2c-(.5*c^2+2(c^3)/6)=1+2c-d
d is a positive value ,slightly less than .008
1+2c=1.24691357...
let us see how this value is affected by subtracting .008 i.e. subtracting more than needed
1.24691357-0.008=1,2389135...while 10c=1.2345678
so sin(c)+cos(c)+ tan(c)>1+2c-d>10c
therefore
sin(c)+cos(c)+ tan(c)>1.2469135....>10c
SO
sin(c)+cos(c)+ tan(c)> 10c
q.e.d.
The above text was corrected and edited following a grave error
spotted by Charlie and broll.
Thanks, guys !
Edited on December 13, 2014, 10:46 am
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