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March 2nd, 2013, 06:33 AM  #1 
Joined: Mar 2013 Posts: 1 Thanks: 0  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? 
March 2nd, 2013, 07:00 AM  #2 
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,867 Thanks: 82 Math Focus: Number Theory  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 LindemannWiestrass theorem, I think. NOTE* : I wrote all of the numbers above in the degree sense. 
March 2nd, 2013, 07:05 AM  #3 
Global Moderator Joined: Dec 2006 Posts: 11,462 Thanks: 300 
As cos(5x) ? 16cos^5(x)  20cos³(x) + 5cos(x) and cos(3x) ? 4cos³(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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