The Maxima on-line user's manual

Algebra Calculator

Trigrat

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.

```          (%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\$
(%i2) bc : sin(a)*sin(3*c)/sin(a+b);
%pi
sin(a) sin(3 (- b - a + ---))
3
(%o2)             -----------------------------
sin(b + a)
(%i3) ba : bc, c=a, a=c;
%pi
sin(3 a) sin(b + a - ---)
3
(%o3)               -------------------------
%pi
sin(a - ---)
3
(%i4) ac2 : ba^2 + bc^2 - 2*bc*ba*cos(b);
2         2         %pi
sin (3 a) sin (b + a - ---)
3
(%o4) ---------------------------
2     %pi
sin (a - ---)
3
%pi
- (2 sin(a) sin(3 a) sin(3 (- b - a + ---)) cos(b)
3
%pi            %pi
sin(b + a - ---))/(sin(a - ---) sin(b + a))
3              3
2       2              %pi
sin (a) sin (3 (- b - a + ---))
3
+ -------------------------------
2
sin (b + a)
(%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```

```(%o1)                                true
(%i2) ```

y:x^2-4;

x:5;

trigrat(y);

Calculate

? trigrat;

Calculate

? trigrat;

Calculate

y:x^2-4;

x:5;

trigrat(y);

Calculate

? trigrat;

Calculate

? trigrat;

Calculate