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March 2nd, 2013, 06:33 AM   #1
 
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Proof, cos is irrational

Hello, I am trying to prove that is an irrational number, but I can't find any clue.
I see a way how to prove irrationality of , but it doesn't help me.

Could you give me any hint, please?
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March 2nd, 2013, 07:00 AM   #2
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Re: Proof, cos is irrational

cos(2) can be expressed in terms of elementary powers of cos(1). If cos(2) is algebraic, so is cos(1). Since cos(20) can be expressed in terms of cos(1) by algebraic means, and cos(20) is irrational, cos(1) must be irrational too. Hence, the proof follows.

EDIT : It can also be proved that cos(1) is transcendental by Lindemann-Wiestrass theorem, I think.

NOTE* : I wrote all of the numbers above in the degree sense.
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March 2nd, 2013, 07:05 AM   #3
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As cos(5x) ? 16cos^5(x) - 20cos(x) + 5cos(x) and cos(3x) ? 4cos(x) - 3cos(x), the rationality of cos(x) implies the rationality of cos(5x) and cos(3x), so if cos(2) were rational, cos(10) and cos(30) would also be rational.

However, cos(30) is ?3/2, which is irrational. Hence cos(2) is irrational.
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