- #1
hadi amiri 4
- 98
- 1
find the numeral value of this
Sin(1)*Sin(2)*Sin(3)*...*Sin(89)
Note:the numbers in brackets are degree
Sin(1)*Sin(2)*Sin(3)*...*Sin(89)
Note:the numbers in brackets are degree
maze said:It's not quite so simple as successive applications of the exact same trick. If you went one further, then you would get pairings that don't quite add up to 90.
86 and 2; 82 and 6; 78 and 10, etc all add to 88
Look at the link yenchin posted. Another way to obtain the result in there is to consider the roots of the equation x^n-1=0, i.e. the nth roots of unity given by 1, z, z^2, ..., z^(n-1), where z=e^(2pi*i/n). This leads us to the equation 1+x+...+x^(n-1)=(x-z)(x-z^2)...(x-z^(n-1)). Plug in x=1 and take absolute values. Finally, try to find a nice expression for |1-z^k|. What do you get?uart said:I got the same 22 term simplification as Kittel Knight. Has anyone found a better solution? Does someone want to post a clue as to the form of the answer we can expect?
morphism said:Look at the link yenchin posted. Another way to obtain the result in there is to consider the roots of the equation x^n-1=0, i.e. the nth roots of unity given by 1, z, z^2, ..., z^(n-1), where z=e^(2pi*i/n). This leads us to the equation 1+x+...+x^(n-1)=(x-z)(x-z^2)...(x-z^(n-1)). Plug in x=1 and take absolute values. Finally, try to find a nice expression for |1-z^k|. What do you get?
uart said:Ok thanks morphism, somehow I missed that link before - very interesting result.
Anyway it follows immediately from that result that the sine product posed in this thread can be simplified to [itex]\sqrt{179/2} \, \times \, 2^{-89} [/itex]
Thanks.
The numeral value of Sin(1)*Sin(2)*Sin(3)* *Sin(89) is approximately 0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000227.
The numeral value was calculated using a scientific calculator or a computer program that can handle very small numbers. The values of the sine function for each angle were multiplied together to get the final result.
This is because the values of the sine function for angles close to 90 degrees become very small. When these small values are multiplied together, the result becomes even smaller.
Yes, the numeral value can be simplified by using trigonometric identities. For example, Sin(90-x) = Cos(x), so the expression can be rewritten as Cos(1)*Cos(2)*Cos(3)* *Cos(89). However, the value will still be very small.
This calculation may have applications in mathematics and physics, where small values are important. It also demonstrates the precision of mathematical functions and the limitations of our numerical systems.