# [seqfan] Re: Two roots of trigonometric polynomials

israel at math.ubc.ca israel at math.ubc.ca
Thu Oct 3 04:04:48 CEST 2013

```Indeed. More generally, cos(mx) = T_m(cos(x)) where T_m is the m'th
Chebyshev polynomial of the first kind, and sin(mx)/sin(x) =
(T_m)'(cos(x))/m

In this case

T_6(c) - T_4(c)^2 = 4 (c^2 - 1) (c^2 + (1-c^2) ((T_3)'(c)/3)^2 - 1/2)

Robert Israel

On Oct 2 2013, allouche at math.jussieu.fr wrote:

>This can also be done "by hand". Putting T = 2 cos(2x),
>we easily have that:
>cos(6x) = (cos(4x))^2 gives
>T^4 - 2 T^3 - 4 T^2 + 6 T + 4 = 0
>and that
>(cos x)^2 + (sin 3x)^2 = 1/2 gives
>T^3 - 4 T - 2 = 0
>but
>T^4 - 2 T^3 - 4 T^2 + 6 T + 4  = (T - 2) (T^3 - 4 T - 2)
>
>jp
>
>
>
>
>
>Charles Greathouse <charles.greathouse at case.edu> a écrit :
>
>> Zero[aa_] :=
>>  With[{a = Exp[2 I*TrigToExp[aa]]},
>>   FullSimplify[Numerator[a] - Denominator[a]] == 0]
>>
>> Zero[(x /.
>>     Solve[Cos[6 x] == Cos[4 x]^2 && 0 < x < 1, x,
>>        Reals][[1]][[1]]) - (x /.
>>     Solve[Cos[x]^2 + Sin[3 x]^2 == 1/2 && 0 < x < 1, x,
>>        Reals][[1]][[1]])]
>>
>> True
>>
>> Charles Greathouse
>> Analyst/Programmer
>> Case Western Reserve University
>>
>>
>> On Wed, Oct 2, 2013 at 2:44 PM, Robert G. Wilson v <rgwv at rgwv.com> wrote:
>>
>>> I also "Played" with Mathematica. I first looked that their graphs
>>> from 0 to Pi. Bob.
>>>
>>> -----Original Message-----
>>> From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of Charles
>>> Greathouse
>>> Sent: Wednesday, October 02, 2013 2:21 PM
>>> To: Sequence Fanatics Discussion list
>>> Subject: [seqfan] Re: Two roots of trigonometric polynomials
>>>
>>> I was able to prove that the constants are the same by the
>>> time-honored method of "playing with Mathematica". I don't yet have
>>> anything nice, but you should be able to recycle A197757 while I look
>>> for something more presentable.
>>>
>>> Charles Greathouse
>>> Analyst/Programmer
>>> Case Western Reserve University
>>>
>>>
>>> On Wed, Oct 2, 2013 at 1:48 PM, Richard J. Mathar
>>> <mathar at mpia-hd.mpg.de>wrote:
>>>
>>> > Is there a way to reduce the trigonometric expressions in
>>> > http://oeis.org/A197757
>>> > and http://oeis.org/A197488 to demonstrate that the two constants are
>>> > the same?
>>> >
>>> > RJM
>>> >
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```