How to prove that cos(10°) is not rational ?

[egou...@gmail.com]

Hello,
I already know that cos(x) is irrational when 0°<x<90° and x<>60° but
I don't find the proof on the Web.
Does someone have a reference for that?

Thx
Erwan

[[Mr.] Lynn Kurtz]

cos(x) varies continuously from 1 to 0 as for 0 <= x <= Pi/2 so it
takes on all values between 0 and 1. Your assertion is false.

--Lynn

[erwan]

> cos(x) varies continuously from 1 to 0 as for 0 <= x <= Pi/2 so it
> takes on all values between 0 and 1. Your assertion is false.
>

Sorry I missed something : cos(x) is irrational when 0°<x<90° and
x<>60° AND x is Rationnal

[Virgil]

In article <1190485694....@o80g2000hse.googlegroups.com>,
egou...@gmail.com wrote:

The reason there is no proof on the web is that what you have stated, at
least in the form in which you have stated it, is false.

Since cos(0º) = 1 and cos(90º) = 0 and cos is continuous, for EVERY
RATIONAL VALUE between 0 and 1 there is an angle between 0º and 90º for
which the cosine takes that rational value.

[erwan]

> The reason there is no proof on the web is that what you have stated, at
> least in the form in which you have stated it, is false.

You didn't read my last statement : cos(x) is irrational when 0°<x<90°
and
x<>60° AND x is Rationnal

For example : cos(10) is irrational

[Ben Rudiak-Gould]

Here's a previous sci.math thread on this subject:

http://groups.google.com/group/sci.math/browse_frm/thread/ebac71b907f1f18c

-- Ben

[Carl Barron]

In article <1190485694....@o80g2000hse.googlegroups.com>,
<egou...@gmail.com> wrote:

> Hello,
> I already know that cos(x) is irrational when 0°<x<90° and x<>60° but

> I don't find the proof on the W6eb.

> Does someone have a reference for that?
>
> Thx
> Erwan
>

cos(3x) = 4cos(x)^3-3cos(x) so if x = cos (10degrees)

cos(30degrees) = sqrt(3)/2 = 4x^3 - 3x

or p(x) = 64x^6 - 96x^4 +36 x^2 - 3 = 0

show this does not have rational roots :)

[Dave L. Renfro]

Erwan wrote:

Below are some references in chronological order.
Some of them give more precise results, such as
which rational degree measure angles give rise
to quadratic irrationals, which give rise to cubic
irrationals, etc.

1868 paper by Hessel (earliest reference I know of)
http://tinyurl.com/375735

R. S. Underwood, American Mathematical Monthly 28 [29]
(1921 [1922]), 374-376 [255-256 + editor's remarks
on pp. 256-257 + 346].

D. H. Lehmer, Amer. Math. Monthly 40 (1933), 165-166.

J. M. H. Olmsted, Amer. Math. Monthly 52 (1945), 507-508.

T. S. Chu, Amer. Math. Monthly 57 (1950), 407-408.

Kenneth W. Wegner, Mathematics Teacher 50 (1957), 557-561.

Kenneth W. Wegner, Amer. Math. Monthly 66 (1959), 52-53.

L. Carlitz and J. M. Thomas, Amer. Math. Monthly
69 (1962), 789-793.

Eugene A. Maier, Amer. Math. Monthly 72 (1965), 1012-1013.

Kenneth W. Wegner, Math. Teacher 60 (1967), 33-37.

H. G. Forder, Mathematical Gazette 53 (1969), 177-178.

Polya/Szego, "Problems and Theorems in Analysis II, 1976.
[I don't know the exact page reference, unfortunately.]

V. Srinivas, Math. Gazette 61 (1977), 290-292.

Desmond MacHale, Math. Gazette 66 (1982), 144-145.

K. Robin McLean, Math. Gazette 67 (1983), 127-128.

D. H. Armitage, Math. Gazette 67 (1983), 128-129.

Desmond MacHale, Math. Gazette 67 (1983), 282-284.

Dave L. Renfro

[Dave L. Renfro]

Erwan wrote:

>> I already know that cos(x) is irrational when 0°<x<90°
>> and x<>60° but I don't find the proof on the Web.
>> Does someone have a reference for that?

Dave L. Renfro wrote (in part):

> Below are some references in chronological order.
> Some of them give more precise results, such as
> which rational degree measure angles give rise
> to quadratic irrationals, which give rise to cubic
> irrationals, etc.

Oops, I forgot the most obvious reference (for those
who have looked into this issue):

Ivan Niven, "Irrational Numbers", Mathematical Association
of America, 1956.

There's quite a bit about this at the end of the second
or third chapter, I forgot which one right now.

Dave L. Renfro

[Dave L. Renfro]

Dave L. Renfro wrote (in part):

> 1868 paper by Hessel (earliest reference I know of)
> http://tinyurl.com/375735

Interestingly, it seems that most (all?) of the
volumes of the journal "Archiv der Mathematik und Physik"
are on the internet via google-books. For example, the
paper I cited can be found at

http://tinyurl.com/253u7f

This is a huge .pdf file to download (28 MB),
and it is very tedious trying to navigate through
its pages as a result. Since I've already gone
to the trouble, if anyone's interested the paper
can be found on pp. 92 to 105 of the .pdf file output
[= journal page numbers 81 to 96]. You'll probably
find it easier to just print these pages than trying
to view them.

Dave L. Renfro