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Groups > comp.soft-sys.math.maple > #1187

Re: Simplify trigonometric expressions

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On 19.08.2015 20:19, Peter Luschny wrote:
> Hi Martin!
>
>> I expect the remainder to be handled in the same manner. But I don't see
>> why Derive should not fail to simplify similar expressions whose
>> trigonometric arguments involve larger denominators, as the rule to
>> handle SIN(3*pi/14) - SIN(pi/14) is not generic.
>
> I include some further examples (array of expressions).
>
> case 7:
>
> [(10/7)*x^4*cos((1/7)*Pi)+(2/7)*cos((1/7)*Pi)+(20/7)*x^2*cos((1/7)*Pi)-(20/7)*x^3*cos((1/7)*Pi)-(10/7)*x*cos((1/7)*Pi)-(20/7)*cos((2/7)*Pi)*x^2-(10/7)*cos((2/7)*Pi)*x^4+(10/7)*x*cos((2/7)*Pi)-(2/7)*cos((2/7)*Pi)+(20/7)*cos((2/7)*Pi)*x^3+(10/7)*x^4*cos((3/7)*Pi)+(2/7)*cos((3/7)*Pi)+(20/7)*x^2*cos((3/7)*Pi)-(20/7)*x^3*cos((3/7)*Pi)-(10/7)*x*cos((3/7)*Pi)-1/7+(5/7)*x-(10/7)*x^2+(10/7)*x^3-(5/7)*x^4+x^5,
> -(6/7)*x+(15/7)*x^2-(20/7)*x^3+(15/7)*x^4-(6/7)*x^5+x^6+(30/7)*cos((2/7)*Pi)*x^4-(12/7)*cos((2/7)*Pi)*x^5-(30/7)*x^4*cos((3/7)*Pi)-(30/7)*x^4*cos((1/7)*Pi)-(2/7)*cos((3/7)*Pi)-(2/7)*cos((1/7)*Pi)-(12/7)*x*cos((2/7)*Pi)+1/7+(2/7)*cos((2/7)*Pi)-(30/7)*x^2*cos((3/7)*Pi)-(30/7)*x^2*cos((1/7)*Pi)+(40/7)*x^3*cos((1/7)*Pi)+(40/7)*x^3*cos((3/7)*Pi)+(12/7)*x*cos((3/7)*Pi)+(12/7)*x*cos((1/7)*Pi)+(12/7)*x^5*cos((3/7)*Pi)+(12/7)*x^5*cos((1/7)*Pi)-(40/7)*cos((2/7)*Pi)*x^3+(30/7)*cos((2/7)*Pi)*x^2];
>
> case 9:
>
> [x+((4/9)*I)*sin((1/9)*Pi)+((4/9)*I)*sin((2/9)*Pi)-(2/9)*cos((2/9)*Pi)-((4/9)*I)*sin((4/9)*Pi)+(2/9)*cos((1/9)*Pi)-(2/9)*cos((4/9)*Pi),
> (4/9)*x*cos((1/9)*Pi)-(2/9)*cos((1/9)*Pi)-(4/9)*cos((2/9)*Pi)*x+(2/9)*cos((2/9)*Pi)-(4/9)*x*cos((4/9)*Pi)+(2/9)*cos((4/9)*Pi)+((8/9)*I)*x*sin((1/9)*Pi)-((8/9)*I)*sin((1/9)*Pi)+((8/9)*I)*x*sin((2/9)*Pi)-((8/9)*I)*sin((2/9)*Pi)+((8/9)*I)*sin((4/9)*Pi)-((8/9)*I)*x*sin((4/9)*Pi)+x^2,
> ((4/3)*I)*sin((2/9)*Pi)*x^2-((8/3)*I)*x*sin((1/9)*Pi)-((8/3)*I)*x*sin((2/9)*Pi)-((4/3)*I)*sin((4/9)*Pi)+((8/3)*I)*x*sin((4/9)*Pi)+((4/3)*I)*x^2*sin((1/9)*Pi)-((4/3)*I)*x^2*sin((4/9)*Pi)+((4/3)*I)*sin((2/9)*Pi)+((4/3)*I)*sin((1/9)*Pi)-(2/3)*x*cos((1/9)*Pi)+(2/3)*x^2*cos((1/9)*Pi)+(2/3)*cos((2/9)*Pi)*x-(2/3)*cos((2/9)*Pi)*x^2+(2/3)*x*cos((4/9)*Pi)-(2/3)*x^2*cos((4/9)*Pi)+x^3,
> ((16/3)*I)*x*sin((1/9)*Pi)+((16/9)*I)*x^3*sin((1/9)*Pi)-((16/3)*I)*x*sin((4/9)*Pi)-((16/3)*I)*x^2*sin((1/9)*Pi)+((16/3)*I)*x^2*sin((4/9)*Pi)-((16/9)*I)*x^3*sin((4/9)*Pi)-((16/3)*I)*sin((2/9)*Pi)*x^2+((16/9)*I)*sin((2/9)*Pi)*x^3+((16/3)*I)*x*sin((2/9)*Pi)+((16/9)*I)*sin((4/9)*Pi)-((16/9)*I)*sin((2/9)*Pi)-((16/9)*I)*sin((1/9)*Pi)+(8/9)*x^3*cos((1/9)*Pi)-(2/9)*cos((1/9)*Pi)-(4/3)*x^2*cos((1/9)*Pi)+(4/3)*cos((2/9)*Pi)*x^2+(2/9)*cos((2/9)*Pi)-(8/9)*cos((2/9)*Pi)*x^3+(4/3)*x^2*cos((4/9)*Pi)+(2/9)*cos((4/9)*Pi)-(8/9)*x^3*cos((4/9)*Pi)+x^4,
> x^5-(10/9)*cos((2/9)*Pi)*x^4+(20/9)*cos((2/9)*Pi)*x^3-(10/9)*x*cos((1/9)*Pi)+(10/9)*x*cos((4/9)*Pi)+(10/9)*cos((2/9)*Pi)*x+(20/9)*x^3*cos((4/9)*Pi)-(20/9)*x^3*cos((1/9)*Pi)-(2/9)*cos((2/9)*Pi)+((20/9)*I)*sin((2/9)*Pi)-((20/9)*I)*sin((4/9)*Pi)+((20/9)*I)*sin((1/9)*Pi)-((80/9)*I)*sin((2/9)*Pi)*x^3+((20/9)*I)*sin((2/9)*Pi)*x^4-((20/9)*I)*x^4*sin((4/9)*Pi)+((20/9)*I)*x^4*sin((1/9)*Pi)+((40/3)*I)*sin((2/9)*Pi)*x^2-((40/3)*I)*x^2*sin((4/9)*Pi)+((40/3)*I)*x^2*sin((1/9)*Pi)+((80/9)*I)*x^3*sin((4/9)*Pi)-((80/9)*I)*x^3*sin((1/9)*Pi)-((80/9)*I)*x*sin((2/9)*Pi)+((80/9)*I)*x*sin((4/9)*Pi)-((80/9)*I)*x*sin((1/9)*Pi)+(10/9)*x^4*cos((1/9)*Pi)-(10/9)*x^4*cos((4/9)*Pi)+(2/9)*cos((1/9)*Pi)-(2/9)*cos((4/9)*Pi)];
>
> case 11:
>
> [-1/11+x+(2/11)*cos((5/11)*Pi)+(2/11)*cos((1/11)*Pi)+(2/11)*cos((3/11)*Pi)-(2/11)*cos((4/11)*Pi)-(2/11)*cos((2/11)*Pi),
> -(2/11)*cos((1/11)*Pi)+(4/11)*x*cos((1/11)*Pi)+(2/11)*cos((2/11)*Pi)-(4/11)*cos((2/11)*Pi)*x-(2/11)*cos((3/11)*Pi)+(4/11)*x*cos((3/11)*Pi)+(2/11)*cos((4/11)*Pi)-(4/11)*x*cos((4/11)*Pi)-(2/11)*cos((5/11)*Pi)+(4/11)*x*cos((5/11)*Pi)+1/11-(2/11)*x+x^2,
> (2/11)*cos((1/11)*Pi)+(6/11)*x^2*cos((1/11)*Pi)-(6/11)*x*cos((1/11)*Pi)-(6/11)*cos((2/11)*Pi)*x^2-(2/11)*cos((2/11)*Pi)+(6/11)*cos((2/11)*Pi)*x+(2/11)*cos((3/11)*Pi)+(6/11)*x^2*cos((3/11)*Pi)-(6/11)*x*cos((3/11)*Pi)-(6/11)*x^2*cos((4/11)*Pi)-(2/11)*cos((4/11)*Pi)+(6/11)*x*cos((4/11)*Pi)+(2/11)*cos((5/11)*Pi)+(6/11)*x^2*cos((5/11)*Pi)-(6/11)*x*cos((5/11)*Pi)-1/11+(3/11)*x-(3/11)*x^2+x^3,
> -(4/11)*x+(6/11)*x^2-(4/11)*x^3+x^4-(8/11)*cos((2/11)*Pi)*x^3+(12/11)*cos((2/11)*Pi)*x^2-(8/11)*cos((2/11)*Pi)*x-(12/11)*x^2*cos((5/11)*Pi)+(12/11)*x^2*cos((4/11)*Pi)-(12/11)*x^2*cos((1/11)*Pi)-(12/11)*x^2*cos((3/11)*Pi)-(2/11)*cos((5/11)*Pi)-(2/11)*cos((1/11)*Pi)-(2/11)*cos((3/11)*Pi)+(2/11)*cos((4/11)*Pi)+(8/11)*x^3*cos((1/11)*Pi)+(8/11)*x^3*cos((3/11)*Pi)+(8/11)*x^3*cos((5/11)*Pi)-(8/11)*x^3*cos((4/11)*Pi)+1/11+(2/11)*cos((2/11)*Pi)-(8/11)*x*cos((4/11)*Pi)+(8/11)*x*cos((3/11)*Pi)+(8/11)*x*cos((1/11)*Pi)+(8/11)*x*cos((5/11)*Pi),
> (5/11)*x-(10/11)*x^2+(10/11)*x^3-(5/11)*x^4+x^5+(20/11)*cos((2/11)*Pi)*x^3-(20/11)*cos((2/11)*Pi)*x^2+(10/11)*cos((2/11)*Pi)*x-(10/11)*cos((2/11)*Pi)*x^4+(20/11)*x^2*cos((5/11)*Pi)-(20/11)*x^2*cos((4/11)*Pi)+(20/11)*x^2*cos((1/11)*Pi)+(20/11)*x^2*cos((3/11)*Pi)+(2/11)*cos((5/11)*Pi)+(2/11)*cos((1/11)*Pi)+(2/11)*cos((3/11)*Pi)-(2/11)*cos((4/11)*Pi)-(20/11)*x^3*cos((1/11)*Pi)-(20/11)*x^3*cos((3/11)*Pi)-(20/11)*x^3*cos((5/11)*Pi)+(20/11)*x^3*cos((4/11)*Pi)-(2/11)*cos((2/11)*Pi)-1/11+(10/11)*x^4*cos((5/11)*Pi)-(10/11)*x^4*cos((4/11)*Pi)+(10/11)*x^4*cos((3/11)*Pi)+(10/11)*x^4*cos((1/11)*Pi)+(10/11)*x*cos((4/11)*Pi)-(10/11)*x*cos((3/11)*Pi)-(10/11)*x*cos((1/11)*Pi)-(10/11)*x*cos((5/11)*Pi)];
>
> Can Rubi handle them?
>
> And: what is the _general_ reduction strategy?
>

Maple does it, using convert(%, RootOf): simplify(%); gives the monomials x^k

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Simplify trigonometric expressions Peter Luschny <peter.luschny@gmail.com> - 2015-08-14 06:00 -0700
  Re: Simplify trigonometric expressions Axel Vogt <&noreply@axelvogt.de> - 2015-08-14 16:02 +0200
    Re: Simplify trigonometric expressions Peter Luschny <peter.luschny@gmail.com> - 2015-08-14 09:37 -0700
    Re: Simplify trigonometric expressions Peter Luschny <peter.luschny@gmail.com> - 2015-08-14 09:45 -0700
      Re: Simplify trigonometric expressions Peter Luschny <peter.luschny@gmail.com> - 2015-08-14 12:39 -0700
      Re: Simplify trigonometric expressions Axel Vogt <&noreply@axelvogt.de> - 2015-08-14 23:45 +0200
        Re: Simplify trigonometric expressions Peter Luschny <peter.luschny@gmail.com> - 2015-08-15 01:43 -0700
        Re: Simplify trigonometric expressions clicliclic@freenet.de - 2015-08-19 14:59 +0200
          Re: Simplify trigonometric expressions Peter Luschny <peter.luschny@gmail.com> - 2015-08-19 11:19 -0700
            Re: Simplify trigonometric expressions Axel Vogt <&noreply@axelvogt.de> - 2015-08-19 22:08 +0200
              Re: Simplify trigonometric expressions Peter Luschny <peter.luschny@gmail.com> - 2015-08-20 06:14 -0700
              Re: Simplify trigonometric expressions clicliclic@freenet.de - 2015-08-22 13:55 +0200

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