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 Trigo inequality (Posted on 2014-12-11)
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 No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
 re(5): solution. Useful (it's to be hoped) table | Comment 9 of 11 |
(In reply to re(4): solution. An idea by Jer)

Let f(x) be sin(x) + cos(x) + tan(x)

`  x       f(x)    f(x)/x0.100   1.19517  11.951720.101   1.19708  11.852250.102   1.19898  11.754720.103   1.20088  11.659070.104   1.20279  11.565250.105   1.20469  11.473210.106   1.20659  11.382900.107   1.20849  11.294270.108   1.21039  11.207280.109   1.21228  11.121870.110   1.21418  11.038000.111   1.21608  10.955640.112   1.21797  10.874740.113   1.21987  10.795270.114   1.22176  10.717180.115   1.22365  10.640440.116   1.22554  10.565020.117   1.22743  10.490880.118   1.22932  10.417990.119   1.23121  10.346320.120   1.23310  10.275830.121   1.23499  10.206510.122   1.23687  10.138310.123   1.23876  10.071210.124   1.24064  10.005190.125   1.24253   9.940220.126   1.24441   9.876270.127   1.24629   9.813330.128   1.24817   9.751360.129   1.25005   9.690340.130   1.25193   9.630260.131   1.25381   9.571080.132   1.25569   9.512800.133   1.25757   9.455390.134   1.25944   9.398830.135   1.26132   9.343090.136   1.26319   9.288180.137   1.26507   9.234060.138   1.26694   9.180710.139   1.26881   9.12813`

When x >= 0.121 (at this level of precision) f(x) exceeds c (Champernowne's number).
When x <= 0.124 f(x) exceeds 10*x.

So there are two ways of coming at it (at least).

 Posted by Charlie on 2014-12-14 08:13:24

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