Cosine , sine , tangent of rational multiples of π
by adityaguharoy, Feb 1, 2017, 5:07 AM
Lemma:
If the polynomial equation
has a non zero rational root 
with 
then 
and 
.
Corollary :
If a polynomial equation has a non zero rational root with and with 
then, 
and .
Theorem :
Let
be a rational mutiple of 
. Then 
, 
, 
are irrational numbers except when 
.
Problem Solving with the above theorem
See here :
https://www.artofproblemsolving.com/community/c140h563991_2013_pumac_team_16
Is
rational ?
If the polynomial equation
has a non zero rational root 
with 
then 
and 
.
by 
and then multiplying by 
we see that 
must be an integer and then using 
we get 
we see that 
must be an integer and then using Corollary :
If a polynomial equation
has a non zero rational root with and with 
then, 
and .Theorem :
Let
be a rational mutiple of 
. Then 
, 
, 
are irrational numbers except when 
.
be any positive integer. First, we prove by induction that there is a polynomial 
of degree 
such that 
for all real numbers 
.
and because of : 
we have 
.
is easily established using elementary trigonometry.
.
is also an integer.
it follows that : 
where the plus sign holds when 
is a solution of 
.
and then applying the above Corollary we conclude that 
and thus 
and thus we get the desired result for 
function.
to get the desired result for 
function.
and then consider the cases when 
function because whenver 
must also be rational and we already know the result for Problem Solving with the above theorem
See here :
https://www.artofproblemsolving.com/community/c140h563991_2013_pumac_team_16
Is
rational ?
is a rational multiple of 
or 
or 
or 
or 
.
be rational ?This post has been edited 5 times. Last edited by adityaguharoy, Feb 13, 2017, 7:07 AM
April 2021
February 2021