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Trigo inequality (Posted on 2014-12-11) Difficulty: 3 of 5
Let c be the Champernowne's constant, or c=0.123456789101112131415161718192021....

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

No Solution Yet Submitted by Danish Ahmed Khan    
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Solution solution | Comment 1 of 11

     

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
  Posted by Ady TZIDON on 2014-12-11 22:27:17

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