My Math Forum Proof, cos is irrational

 Algebra Pre-Algebra and Basic Algebra Math Forum

 March 2nd, 2013, 05:33 AM #1 Newbie   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, 06:00 AM #2 Math Team     Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 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, 06:05 AM #3 Global Moderator   Joined: Dec 2006 Posts: 18,053 Thanks: 1395 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.

 Tags cos, irrational, proof

,

,

,

,

,

,

,

,

,

,

,

# cos1 is irrational

Click on a term to search for related topics.
 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post Chimpeyes Applied Math 14 November 26th, 2013 05:20 AM lun123 Real Analysis 3 October 7th, 2013 08:57 AM Stuck Man Number Theory 1 February 10th, 2012 04:44 AM jstarks4444 Number Theory 1 May 5th, 2011 05:27 PM jstarks4444 Number Theory 4 November 17th, 2010 06:59 AM

 Contact - Home - Forums - Cryptocurrency Forum - Top