monotonic decrease of sin (revivalist version)

[vdmcc]

Neil,
for what it's worth :
----------------------------------------------------------------------------
---
Sin[n] decreases monotonically to -1 :
z={};current=1;
Timing[Do[If[Sin[n]<current,AppendTo[z,current=Sin[n]]],{n, 30000}]]

{1,3,4,5,11,344,1054,1764,2474,3184,3894,4604,5314,6024,6734,7444,8154,8864,
  9574,10284,10994,11704,12414,13124,13834,14544,15254,15964,16674,17384,
  18094,18804,19514,20224,20934,21644,22354,23064,23774,24484,25194,25904,
  26614,27324,28034,28744,29454}
is a dumb series, because the differences are
Rest[% ]-Drop[% ,-1]

{2,1,1,6,333,710,710,710...
----------------------------------------------------------------------------
---
Abs[Sin[n]] decreases monotonically to 0 :
z={};current=1;
Timing[Do[
    If[Abs[Sin[n]]<current,AppendTo[z,current=Abs[Sin[n]]]],{n,105000}]]

{Sin[1],Sin[3],-Sin[22],-Sin[333],-Sin[355],-Sin[103993],-Sin[104348]}
and probably the numerators of CF[Pi] appear, suggesting a next term of
208341.
----------------------------------------------------------------------------
---
Sin[n] decreases monotonically to 0 :

z={};current=1;
Timing[Do[
    If[If[(t=Sin[n])>0,t,1]<current,AppendTo[z,current=t ]],{n,105000 }]]

{Sin[1],Sin[3],Sin[44],Sin[311],Sin[377],Sin[688],Sin[710],Sin[103638],
  Sin[104703]}
----------------------------------------------------------------------------
---
Cos[n] decreases monotonically to -1 :
z={};current=1;
Timing[Do[If[Cos[n]<current,AppendTo[z,current=Cos[n]]],{n,105000}]]

{Cos[1],Cos[2],Cos[3],Cos[22],Cos[355],Cos[104348]}
----------------------------------------------------------------------------
---
Abs[Cos[n]] decreases monotonically to 0 :
{1,-2,5,-8,11,-344,699,-1054,1409,-1764,2119,-2474,2829,-3184,3539,-3894,
  4249,-4604,4959,-5314,5669,-6024,6379,-6734,7089,-7444,7799,-8154,
  8509,-8864,9219,-9574,9929,-10284,10639,-10994,11349,-11704,12059,-12414,
  12769,-13124,13479,-13834,14189,-14544,14899,-15254,15609,-15964,
  16319,-16674,17029,-17384,17739,-18094,18449,-18804,19159,-19514,
  19869,-20224,20579,-20934,21289,-21644,21999,-22354,22709,-23064,
  23419,-23774,24129,-24484,24839,-25194,25549,-25904,26259,-26614,
  26969,-27324,27679,-28034,28389,-28744,29099,-29454,29809,-30164,
  30519,-30874,31229,-31584,31939,-32294,32649,-33004,33359,-33714,
  34069,-34424,34779,-35134,35489,-35844,36199,-36554,36909,-37264,
  37619,-37974,38329,-38684,39039,-39394,39749,-40104,40459,-40814,
  41169,-41524,41879,-42234,42589,-42944,43299,-43654,44009,-44364,
  44719,-45074,45429,-45784,46139,-46494,46849,-47204,47559,-47914,
  48269,-48624,48979,-49334,49689,-50044,50399,-50754,51109,-51464,51819,
  52174}
is again dumb, the differences (of the unsigned arguments) being
{1,3,3,3,333,355,355,355,355,355...
Oh yes, Mathematica writes Abs[Cos[2]] as -Cos[2], hence the signs.
----------------------------------------------------------------------------
---
Abs[Tan[n]] decreases monotonically to 0 :
z={};current=1;
Timing[Do[
    If[Abs[Tan[n]]<current,AppendTo[z,current=Abs[Tan[n]]]],{n,105000}]]
{-Tan[3],Tan[22],-Tan[333],Tan[355],-Tan[103993],Tan[104348]}
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---

So let's be carefull with Gerards early generalisations (;-))

wouter




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