Show sin(1), in degrees, is algebraic over the rationals

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SUMMARY

The discussion centers on demonstrating that sin(1 degree) is algebraic over the rationals by finding its minimal polynomial. The user attempts to express sin(1 degree) as sin(pi/180) and references the known identity sin(pi/5) = (1/4)*√{10 - 2√5}. Additionally, the discussion includes the exponential form of sine, sin(z) = (e^(iz) - e^(-iz))/(2i), as a method to explore the algebraic properties of sine functions.

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Homework Statement



show sin(1), in degrees, is algebraic over the rationals

The Attempt at a Solution



I want to find the minimal polynomial that has root sin(pi/180). Ya that's all I got.
 
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Well, here's one way.

sin(pi/5) = (1/4)*\sqrt{ 10 - 2*\sqrt{5} }


Here's another.

sin z = ( e^{iz} - e^{-iz} )/(2i)
 

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