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SciMax Toolbox >> trigrat

trigrat

Maxima Function

Calling Sequence

trigrat (expr)

Description

Gives a canonical simplifyed quasilinear form of a trigonometrical expression; expr is a rational fraction of several sin, cos or tan, the arguments of them are linear forms in some variables (or kernels) and %pi/n (n integer) with integer coefficients. The result is a simplified fraction with numerator and denominator linear in sin and cos. Thus trigrat linearize always when it is possible.

(%i1) trigrat(sin(3*a)/sin(a+%pi/3));
(%o1)            sqrt(3) sin(2 a) + cos(2 a) - 1

The following example is taken from Davenport, Siret, and Tournier, Calcul Formel, Masson (or in English, Addison-Wesley), section 1.5.5, Morley theorem.

(%i1) c: %pi/3 - a - b;
                                    %pi
(%o1)                     - b - a + ---
                                     3
(%i2) bc: sin(a)*sin(3*c)/sin(a+b);
                      sin(a) sin(3 b + 3 a)
(%o2)                 ---------------------
                           sin(b + a)
(%i3) ba: bc, c=a, a=c$
(%i4) ac2: ba^2 + bc^2 - 2*bc*ba*cos(b);
         2       2
      sin (a) sin (3 b + 3 a)
(%o4) -----------------------
               2
            sin (b + a)
                                        %pi
   2 sin(a) sin(3 a) cos(b) sin(b + a - ---) sin(3 b + 3 a)
                                         3
 - --------------------------------------------------------
                           %pi
                   sin(a - ---) sin(b + a)
                            3
      2         2         %pi
   sin (3 a) sin (b + a - ---)
                           3
 + ---------------------------
             2     %pi
          sin (a - ---)
                    3
(%i5) trigrat (ac2);
(%o5) - (sqrt(3) sin(4 b + 4 a) - cos(4 b + 4 a)
 - 2 sqrt(3) sin(4 b + 2 a) + 2 cos(4 b + 2 a)
 - 2 sqrt(3) sin(2 b + 4 a) + 2 cos(2 b + 4 a)
 + 4 sqrt(3) sin(2 b + 2 a) - 8 cos(2 b + 2 a) - 4 cos(2 b - 2 a)
 + sqrt(3) sin(4 b) - cos(4 b) - 2 sqrt(3) sin(2 b) + 10 cos(2 b)
 + sqrt(3) sin(4 a) - cos(4 a) - 2 sqrt(3) sin(2 a) + 10 cos(2 a)
 - 9)/4
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