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 Trigonometric product (Posted on 2013-12-29)
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 No Rating

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 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   nextOK  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      0.45464871OK`

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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