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Groups > comp.soft-sys.math.maple > #1189

Re: Simplify trigonometric expressions

Message-ID <55D86322.5E95C3E8@freenet.de> (permalink)
Date 2015-08-22 13:55 +0200
From clicliclic@freenet.de
Newsgroups comp.soft-sys.math.maple, sci.math.symbolic
Subject Re: Simplify trigonometric expressions
References (2 earlier) <ae0050b5-53a8-4965-9c1d-4ed536935efd@googlegroups.com> <d375seF7ivpU1@mid.individual.net> <55D47DAC.19F9EDD3@freenet.de> <9fad8e72-8881-422b-ba90-715fb3544c25@googlegroups.com> <d3k62rFg5urU1@mid.individual.net>
Organization synergetic AG

Cross-posted to 2 groups.

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Axel Vogt schrieb:
> 
> On 19.08.2015 20:19, Peter Luschny wrote:
> >
> >> 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:
> >
> > [...];
> >
> > case 9:
> >
> > [...];
> >
> > case 11:
> >
> > [...];
> >
> > Can Rubi handle them?
> >
> > And: what is the _general_ reduction strategy?
> >
> 
> Maple does it, using convert(%, RootOf): simplify(%); gives the
> monomials x^k

Automatic simplification on Derive 6.10 (not Rubi!) reduces Peter's
vector expressions as follows:

[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)-2~
0/7*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)+1~
2/7*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]

[x^5,x^6]

[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)*S~
IN(1/9*pi)+8/9*x^3*COS(1/9*pi)-2/9*COS(1/9*pi)-4/3*x^2*COS(1/9*p~
i)+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*C~
OS(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)*SI~
N(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)]

[x,x^2,x^3,x^4,x^5]

[-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)+8/1~
1*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*CO~
S(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*p~
i)-10/11*x*COS(3/11*pi)-10/11*x*COS(1/11*pi)-10/11*x*COS(5/11*pi~
)]

[-2*COS(2*pi/11)/11+2*COS(pi/11)/11+2*SIN(5*pi/22)/11-2*SIN(3*pi~
/22)/11+2*SIN(pi/22)/11+x-1/11,(2/11-4*x/11)*COS(2*pi/11)+(4*x/1~
1-2/11)*COS(pi/11)+(4*x/11-2/11)*SIN(5*pi/22)+(2/11-4*x/11)*SIN(~
3*pi/22)+(4*x/11-2/11)*SIN(pi/22)+x^2-2*x/11+1/11,-(6*x^2/11-6*x~
/11+2/11)*COS(2*pi/11)+(6*x^2/11-6*x/11+2/11)*COS(pi/11)+(6*x^2/~
11-6*x/11+2/11)*SIN(5*pi/22)-(6*x^2/11-6*x/11+2/11)*SIN(3*pi/22)~
+(6*x^2/11-6*x/11+2/11)*SIN(pi/22)+x^3-3*x^2/11+3*x/11-1/11,-(8*~
x^3/11-12*x^2/11+8*x/11-2/11)*COS(2*pi/11)+(8*x^3/11-12*x^2/11+8~
*x/11-2/11)*COS(pi/11)+(8*x^3/11-12*x^2/11+8*x/11-2/11)*SIN(5*pi~
/22)-(8*x^3/11-12*x^2/11+8*x/11-2/11)*SIN(3*pi/22)+(8*x^3/11-12*~
x^2/11+8*x/11-2/11)*SIN(pi/22)+x^4-4*x^3/11+6*x^2/11-4*x/11+1/11~
,-(10*x^4/11-20*x^3/11+20*x^2/11-10*x/11+2/11)*COS(2*pi/11)+(10*~
x^4/11-20*x^3/11+20*x^2/11-10*x/11+2/11)*COS(pi/11)+(10*x^4/11-2~
0*x^3/11+20*x^2/11-10*x/11+2/11)*SIN(5*pi/22)-(10*x^4/11-20*x^3/~
11+20*x^2/11-10*x/11+2/11)*SIN(3*pi/22)+(10*x^4/11-20*x^3/11+20*~
x^2/11-10*x/11+2/11)*SIN(pi/22)+x^5-5*x^4/11+10*x^3/11-10*x^2/11~
+5*x/11-1/11]

So, as anticipated, Derive's rule set cannot fully handle the case of
argument denominator 11. Here are the reduction steps for the middle
element with argument denominator 9:

(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

" SIN(n*pi) -> COS((1/2-n)*pi) "

8*#i*x*SIN(4*pi/9)/3+4*#i*x^2*SIN(pi/9)/3-4*#i*x^2*SIN(4*pi/9)/3~
+4*#i*SIN(2*pi/9)/3+4*#i*SIN(pi/9)/3-2*x*COS(pi/9)/3+2*x^2*COS(p~
i/9)/3+2*x*COS(2*pi/9)/3-2*x^2*COS(2*pi/9)/3+2*x*COS(4*pi/9)/3-2~
*x^2*COS(4*pi/9)/3-4*#i*COS(pi/18)/3+#i*((4*x^2/3-8*x/3)*SIN(2*p~
i/9)-8*x*SIN(pi/9)/3)+x^3

" SIN(n*pi) -> COS((1/2-n)*pi) "

8*#i*x*COS(pi/18)/3+4*#i*x^2*SIN(pi/9)/3-4*#i*x^2*SIN(4*pi/9)/3+~
4*#i*SIN(2*pi/9)/3+4*#i*SIN(pi/9)/3-2*x*COS(pi/9)/3+2*x^2*COS(pi~
/9)/3+2*x*COS(2*pi/9)/3-2*x^2*COS(2*pi/9)/3+2*x*COS(4*pi/9)/3-2*~
x^2*COS(4*pi/9)/3-4*#i*COS(pi/18)/3+#i*((4*x^2/3-8*x/3)*SIN(2*pi~
/9)-8*x*SIN(pi/9)/3)+x^3

" SIN(n*pi) -> COS((1/2-n)*pi) "

-4*#i*x^2*COS(pi/18)/3+#i*(8*x*COS(pi/18)/3+4*x^2*SIN(pi/9)/3)+4~
*#i*SIN(2*pi/9)/3+4*#i*SIN(pi/9)/3-2*x*COS(pi/9)/3+2*x^2*COS(pi/~
9)/3+2*x*COS(2*pi/9)/3-2*x^2*COS(2*pi/9)/3+2*x*COS(4*pi/9)/3-2*x~
^2*COS(4*pi/9)/3-4*#i*COS(pi/18)/3+#i*((4*x^2/3-8*x/3)*SIN(2*pi/~
9)-8*x*SIN(pi/9)/3)+x^3

" SIN(z)+SIN(w) -> 2*SIN(z/2+w/2)*COS(z/2-w/2) "

#i*(4*x^2*SIN(pi/9)/3+(8*x/3-4*x^2/3)*COS(pi/18)+8*SIN(pi/6)*COS~
(pi/18)/3)-2*x*COS(pi/9)/3+2*x^2*COS(pi/9)/3+2*x*COS(2*pi/9)/3-2~
*x^2*COS(2*pi/9)/3+2*x*COS(4*pi/9)/3-2*x^2*COS(4*pi/9)/3-4*#i*CO~
S(pi/18)/3+#i*((4*x^2/3-8*x/3)*SIN(2*pi/9)-8*x*SIN(pi/9)/3)+x^3

" SIN(pi/6) -> 1/2 "

#i*(4*x^2*SIN(pi/9)/3+(8*x/3-4*x^2/3)*COS(pi/18)+4*COS(pi/18)/3)~
-2*x*COS(pi/9)/3+2*x^2*COS(pi/9)/3+2*x*COS(2*pi/9)/3-2*x^2*COS(2~
*pi/9)/3+2*x*COS(4*pi/9)/3-2*x^2*COS(4*pi/9)/3-4*#i*COS(pi/18)/3~
+#i*((4*x^2/3-8*x/3)*SIN(2*pi/9)-8*x*SIN(pi/9)/3)+x^3

" COS(z)-COS(w) -> -2*SIN(z/2-w/2)*SIN(z/2+w/2) "

2*(-2*x^2/3+2*x/3)*SIN(pi/18)*COS(2*pi/3)+#i*(4*x^2*SIN(pi/9)/3-~
(4*x^2/3-8*x/3-4/3)*COS(pi/18))+2*x*COS(4*pi/9)/3-2*x^2*COS(4*pi~
/9)/3-4*#i*COS(pi/18)/3+#i*((4*x^2/3-8*x/3)*SIN(2*pi/9)-8*x*SIN(~
pi/9)/3)+x^3

" COS(n*pi) -> SIN((1/2-n)*pi) "

-4*x*(1-x)*SIN(pi/6)*SIN(pi/18)/3+#i*(4*x^2*SIN(pi/9)/3-(4*x^2/3~
-8*x/3-4/3)*COS(pi/18))+2*x*COS(4*pi/9)/3-2*x^2*COS(4*pi/9)/3-4*~
#i*COS(pi/18)/3+#i*((4*x^2/3-8*x/3)*SIN(2*pi/9)-8*x*SIN(pi/9)/3)~
+x^3

" SIN(pi/6) -> 1/2 "

2*x*(x-1)*SIN(pi/18)/3+#i*(4*x^2*SIN(pi/9)/3-(4*x^2/3-8*x/3-4/3)~
*COS(pi/18))+2*x*COS(4*pi/9)/3-2*x^2*COS(4*pi/9)/3-4*#i*COS(pi/1~
8)/3+#i*((4*x^2/3-8*x/3)*SIN(2*pi/9)-8*x*SIN(pi/9)/3)+x^3

" COS(n*pi) -> SIN((1/2-n)*pi) "

2*x*(x-1)*SIN(pi/18)/3+2*x*SIN(pi/18)/3+#i*(4*x^2*SIN(pi/9)/3-4*~
(x^2-2*x-1)*COS(pi/18)/3)-2*x^2*COS(4*pi/9)/3-4*#i*COS(pi/18)/3+~
#i*((4*x^2/3-8*x/3)*SIN(2*pi/9)-8*x*SIN(pi/9)/3)+x^3

" COS(n*pi) -> SIN((1/2-n)*pi) "

2*x^2*SIN(pi/18)/3-2*x^2*SIN(pi/18)/3+#i*(4*x^2*SIN(pi/9)/3-4*(x~
^2-2*x-1)*COS(pi/18)/3)-4*#i*COS(pi/18)/3+#i*((4*x^2/3-8*x/3)*SI~
N(2*pi/9)-8*x*SIN(pi/9)/3)+x^3

" SIN(z)+SIN(w) -> 2*SIN(z/2+w/2)*COS(z/2-w/2) "

#i*(2*(4*x^2/3-8*x/3)*SIN(pi/6)*COS(pi/18)-4*x*(x-2)*COS(pi/18)/~
3)+x^3

" SIN(pi/6) -> 1/2 "

#i*(4*x*(x-2)*COS(pi/18)/3+4*x*(2-x)*COS(pi/18)/3)+x^3

" one final step "

x^3

Examples of Derive's rule SIN(3*pi/14) - SIN(pi/14) -> COS(pi/7) - 1/2
extended to higher denominators are SIN(5*pi/22) - SIN(3*pi/22) +
SIN(pi/22) -> COS(2*pi/11) - COS(pi/11) + 1/2 and SIN(5*pi/26) -
SIN(3*pi/26) + SIN(pi/26) -> COS(3*pi/13) - COS(2*pi/13) + COS(pi/13) -
1/2. Generalization to arbitrary denominators seems straightforward.
Numerical approximation to high precision could be an acceptable
alternative in many situations.

Martin.

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Thread

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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