Maxima Function
trigrat (expr)
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.
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