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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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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.
Last edited by mathbalarka
on Sat Mar 02, 2013 10:08 am, edited 1 time in total.
Carl Friedrich Gauss —
When a philosopher says something that is true then it is trivial.
When he says something that is not trivial then it is false.
Pierre-Simon Laplace —
[said about Napier's logarithms:] ...by shortening the labours doubled the life of the astronomer.
Henri Poincaré —
Les faits ne parlent pas (Facts do not speak).Advanced Analysis Of Quintics
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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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