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Re: How to simplify certain solutions of trigonometric equation?

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Le dimanche 8 mars 2015 15:22:56 UTC-4, Rainer Rosenthal a écrit :
> I was searching for solutions of the following equation:
> 
>   sin(x) + sin(2x) + sin(3x) = cos(x) + cos(2x) + cos(3x)
> 
> So I defined funtion f:
> f := x -> sin(x) + sin(2*x) + sin(3*x) - (cos(x) + cos(2*x) + cos(3*x));
> 
> and solved as follows:
> p := [solve(f(x)=0,x)];
> 
> I got six different solutions:
> 
> p[1] = 2/3*Pi
> p[2] = -2/3*Pi
> p[3] = arctan(2*(1/2*(2+sqrt(2))^(3/2)-3/2*sqrt(2+sqrt(2)))/(sqrt(2+sqrt(2))))
> p[4] = -arctan(2*(-1/2*(2+sqrt(2))^(3/2)+3/2*sqrt(2+sqrt(2)))/(sqrt(2+sqrt(2))))-Pi
> p[5] = arctan(2*(1/2*(2-sqrt(2))^(3/2)-3/2*sqrt(2-sqrt(2)))/(sqrt(2-sqrt(2))))
> p[6] = -arctan(2*(-1/2*(2-sqrt(2))^(3/2)+3/2*sqrt(2-sqrt(2)))/(sqrt(2-sqrt(2))))+Pi]
> 
> I hoped for a simple zero for each f(p[i]), but most of the time I got
> difficult expressions, which were only simplified by "simplify":
> 
> I usd these Maple-lines:
> for i to nops(p) do   print(Check_Solution_Number_,i);
>    simplify(f(p[i]));
> od;
> 
> and received the following output:
> 
> 
>                       Check_Solution_Number_, 1
> 
> 
>                                   0
> 
> 
>                       Check_Solution_Number_, 2
> 
> 
>                                   0
> 
> 
>                       Check_Solution_Number_, 3
> 
> 
>   (-2 + sqrt(2) + sin(3/8 Pi) sqrt(4 - 2 sqrt(2))
> 
>          - cos(3/8 Pi) sqrt(4 - 2 sqrt(2)))/sqrt(4 - 2 sqrt(2))
> 
> 
>                       Check_Solution_Number_, 4
> 
> 
>   (2 - sqrt(2) - sin(2 %2) %1 + sin(3 %2) %1 - cos(2 %2) %1
> 
>          + cos(3 %2) %1)/sqrt(4 - 2 sqrt(2))
> 
>   %1 := sqrt(4 - 2 sqrt(2))
> 
>   %2 := arctan(-sqrt(2) + 1)
> 
> 
>                       Check_Solution_Number_, 5
> 
> 
>   - (2 + sqrt(2) - sin(2 %2) %1 - sin(3 %2) %1 + cos(2 %2) %1
> 
>          + cos(3 %2) %1)/sqrt(4 + 2 sqrt(2))
> 
>   %1 := sqrt(4 + 2 sqrt(2))
> 
>   %2 := arctan(-1 - sqrt(2))
> 
> 
>                       Check_Solution_Number_, 6
> 
> 
>   (2 + sqrt(2) - sin(1/8 Pi) sqrt(4 + 2 sqrt(2))
> 
>          - cos(1/8 Pi) sqrt(4 + 2 sqrt(2)))/sqrt(4 + 2 sqrt(2))
> 
> 
> Which options of "simplify" could do better?
> (I still use Maple V).
> 
> Cheers,
> Rainer


Try something like this,

  seq( radnormal(convert(radnormal(f(t)),radical)), t=p );

or

  seq( simplify(convert(simplify(f(t)),radical)), t=p );

because it looks like your MapleV is missing the step of turning certain trig constants into their radical equivalents.

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Thread

How to simplify certain solutions of trigonometric equation? Rainer Rosenthal <r.rosenthal@web.de> - 2015-03-08 20:25 +0100
  Re: How to simplify certain solutions of trigonometric equation? acer <maple@rogers.com> - 2015-03-08 18:48 -0700
    Re: How to simplify certain solutions of trigonometric equation? Rainer Rosenthal <r.rosenthal@web.de> - 2015-03-09 11:05 +0100
      Re: How to simplify certain solutions of trigonometric equation? acer <maple@rogers.com> - 2015-03-09 06:09 -0700
        Re: How to simplify certain solutions of trigonometric equation? Rainer Rosenthal <r.rosenthal@web.de> - 2015-03-09 14:40 +0100
          Re: How to simplify certain solutions of trigonometric equation? acer <maple@rogers.com> - 2015-03-09 07:36 -0700
            Re: How to simplify certain solutions of trigonometric equation? acer <maple@rogers.com> - 2015-03-09 07:48 -0700
            Re: How to simplify certain solutions of trigonometric equation? Rainer Rosenthal <r.rosenthal@web.de> - 2015-03-09 17:56 +0100
              Re: How to simplify certain solutions of trigonometric equation? acer <maple@rogers.com> - 2015-03-09 10:15 -0700
                Re: How to simplify certain solutions of trigonometric equation? Rainer Rosenthal <r.rosenthal@web.de> - 2015-03-10 01:12 +0100

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