Sin Values of 87 and 89 Degrees

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Discussion Overview

The discussion revolves around the value of the product $\sin (1^0).\sin (3^0).\sin (5^0)...\sin (87^0).\sin (89^0)$, with all angles expressed in degrees. Participants explore various methods to compute this value, including algebraic and complex number approaches.

Discussion Character

  • Exploratory
  • Mathematical reasoning

Main Points Raised

  • One participant states the value of the product is approximately \(4.0194366942304562\times 10^{-14}\), but does not provide a method for deriving this result algebraically.
  • Another participant expresses interest in finding the value using complex numbers, specifically mentioning the use of nth-roots of unity.
  • A different participant references a method from an external link, suggesting that the angles involved are solutions to the equation $\cos(90x) = 0$.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the method to solve the problem, and multiple approaches are discussed without agreement on a definitive solution.

Contextual Notes

There are limitations in the discussion regarding the lack of detailed algebraic methods and the dependence on external resources for potential solutions.

juantheron
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The value of $\sin (1^0).\sin (3^0).\sin (5^0)...\sin (87^0).\sin (89^0)$

where all angles are in degree
 
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jacks said:
The value of $\sin (1^0).\sin (3^0).\sin (5^0)...\sin (87^0).\sin (89^0)$

where all angles are in degree

Hi jacks,

I haven't found a way solve this algebraically. But if you are interested about the answer it is, \(4.0194366942304562\times 10^{-14}\)
 
Sudharaka said:
Hi jacks,

I haven't found a way solve this algebraically. But if you are interested about the answer it is, \(4.0194366942304562\times 10^{-14}\)

Thanks Sudhakara

I am trying to find it with the help of complex no.(like nth -roots of unity)
 
jacks said:
The value of $\sin (1^0).\sin (3^0).\sin (5^0)...\sin (87^0).\sin (89^0)$

where all angles are in degree
Follow the method used in http://www.mathhelpboards.com/showthread.php?253-Simplify-cos(a)cos(2a)cos(3a)-cos(999a)-if-a-(2pi)-1999&p=1517&viewfull=1#post1517, noting that $x=\pm1^\circ,\pm3^\circ,\pm5^\circ,\ldots,\pm89 ^\circ$ are the solutions of the equation $\cos(90x) = 0.$
 

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