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Trigonometric product (Posted on 2013-12-29) Difficulty: 2 of 5
Compute the infinite product

[sin(x) cos(x/2)]^1/2 [sin(x/2) cos(x/4)]^1/4 [sin(x/4) cos(x/8)]^1/8 ... ,

where 0 ≤ x ≤ 2π

No Solution Yet Submitted by Danish Ahmed Khan    
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Solution computer solution | Comment 2 of 4 |

After modifying the program to use degree input converted to radian and finding that 90 resulted in a value of .5, my first thought was the formula gave sin(x/3), but that didn't pan out for other values. The next thought was sin(x)/2. That did work:

list
   10   for Xn=1 to 20:X=Xn/10
   20   Deno=1:Deno2=2:A=1
   30   for F=1 to 50
   40    Fctr=(sin(X/Deno)*cos(X/(Deno2)))^(1/(Deno2))
   50    A=A*Fctr
   70    Deno=Deno2:Deno2=Deno2*2
   80   next
   90   print X,using(1,8),A,asin(A),sin(X)/2
  100   next
OK

  x     inf. product     arcsin(prod)    sin(x) / 2
 0.1     0.04991671      0.04993746      0.04991671
 0.2     0.09933467      0.09949876      0.09933467
 0.3     0.14776010      0.14830313      0.14776010
 0.4     0.19470917      0.19596094      0.19470917
 0.5     0.23971277      0.24206998      0.23971277
 0.6     0.28232124      0.28621292      0.28232124
 0.7     0.32210884      0.32795621      0.32210884
 0.8     0.35867805      0.36685132      0.35867805
 0.9     0.39166345      0.40243879      0.39166345
 1.0     0.42073549      0.43425591      0.42073549
 1.1     0.44560368      0.46184848      0.44560368
 1.2     0.46601954      0.48478658      0.46601954
 1.3     0.48177909      0.50268383      0.48177909
 1.4     0.49272486      0.51521835      0.49272486
 1.5     0.49874749      0.52215311      0.49874749
 1.6     0.49978680      0.52335261      0.49978680
 1.7     0.49583241      0.51879310      0.49583241
 1.8     0.48692382      0.50856438      0.48692382
 1.9     0.47315004      0.49286296      0.47315004
 2.0     0.45464871      0.47197777      0.45464871
OK

The infinite product column and the sin(x)/2 column match.


  Posted by Charlie on 2013-12-29 14:56:59
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