Re: When is cos(x) rational?
- From: Spamless <Spamless@xxxxxxx>
- Date: 05 Mar 2006 03:57:41 GMT
On 2006-03-03, =?ISO-8859-1?Q?Jos=E9_Carlos_Santos?= <jcsantos@xxxxxxxx> wrote:
Hi all:
I computed the angles between certain pairs of vectors in
3-dimensional space. For instance, if V and V' are two vertices of a
regular tetrahedron centered at C, the the cosine of the angle VCV'
is 1/3.
Should that not be -1/3?
(center (0,0,0): vertices (1,1,1),(1,-1,-1),(-1,-1,1),(-1,1,-1))
My first reaction was that then the angle is not a rational
multiple of pi. However, the truth is that I am not sure about that.
So, my question is: are 0, 1/3, and 1/2 the only rational numbers
_r_ in [0,1/2] such that cos(r*pi) is rational?
I like to think of it ... for what theta=m*2*pi/n is the tan^2(theta)
rational (it seems to combine cases - if the sine or cosine is rational so
then is the tangent, squared). Write z=exp(i*m*2*pi/n) and writing
tan(theta) in terms of z one sees that IF tan^2(theta) is rational then z
satisfies a fourth order equation with rational coefficients. As theta is a
rational multiple of pi, z is a root of one and so its mimimal polynomial
over Q is a cyclotomic polynomial with order phi(n) and as z satisfies some
equation of degree four, one must have phi(n)<=4. There are not too many
cases to consider.
The only angles which are a rational number of degrees (the same as a
rational multiple of pi) and which could possibly have a sine, tangent or
cosine which is rational (in fact which could possibly have a sine and
cosine whose squares are rational) must have phi(n)<=4.
Checking the angles in the first quadrant ...
0_degrees (sine, cosine, tangent rational)
30_degrees (sine rational)
45_degrees ( tangent rational)
60_degrees ( cosine rational)
90_degrees (sine, cosine rational)
(n=12, phi(n)=4 for 30_degrees
n= 8, phi(n)=4 for 45_degrees
n= 6, phi(n)=2 for 60_degrees
n= 4, phi(n)=2 for 90_degrees)
.
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