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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 $\cos(2^{\circ})$ is an irrational number, but I can't find any clue. I see a way how to prove irrationality of $\tan(2^{\circ})$, 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: 3,871 Thanks: 85 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 Lindemann-Wiestrass 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: 12,927 Thanks: 654 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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### prove that cos1 is irrational number

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