
Algebra PreAlgebra and Basic Algebra Math Forum 
 LinkBack  Thread Tools  Display Modes 
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: 3,871 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 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: 12,364 Thanks: 527 
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 
Search tags for this page 
Irrationality of cos(x),prove cos 1 is irrational,Cos10 rational or irrational,prove that cos x is irrational,cos5x=16cos x_20cos x 5cos x
Click on a term to search for related topics.

Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Proof of irrational numbers? Need Feedback.  Chimpeyes  Applied Math  14  November 26th, 2013 06:20 AM 
simply rational/irrational proof  lun123  Real Analysis  3  October 7th, 2013 08:57 AM 
Quadratic irrational proof  Stuck Man  Number Theory  1  February 10th, 2012 05:44 AM 
Irrational Existence Proof  jstarks4444  Number Theory  1  May 5th, 2011 05:27 PM 
Irrational Uncountability proof  jstarks4444  Number Theory  4  November 17th, 2010 07:59 AM 