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From: Rainer Rosenthal <r.rosenthal@web.de>
Newsgroups: comp.soft-sys.math.maple
Subject: How to simplify certain solutions of trigonometric equation?
Date: Sun, 08 Mar 2015 20:25:49 +0100
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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