96.2.4 Factor Sums of sin and cos Functions

Function: trigfactor (x) ¶

The function trigfactor factors expresions of form \(\pm \sin x \pm \cos y.\)

Examples:

1.  

   (%i2) trigfactor(sin(x)+cos(x));
                                               %pi
   (%o2)                       sqrt(2) cos(x - ---)
                                                4
   (%i3) trigrat(%);
   (%o3)                          sin(x) + cos(x)
   

2.  

   (%i4) trigfactor(sin(x)+cos(y));
                              y   x   %pi      y   x   %pi
   (%o4)                2 cos(- - - + ---) cos(- + - - ---)
                              2   2    4       2   2    4
   

   (%i5) trigrat(%);
   (%o5)                          cos(y) + sin(x)
   

3.  

   (%i6) trigfactor(sin(x)-cos(3*y));
                            3 y   x   %pi      3 y   x   %pi
   (%o6)              2 sin(--- - - + ---) sin(--- + - - ---)
                             2    2    4        2    2    4
   (%i7) trigrat(%);
   (%o7)                         sin(x) - cos(3 y)
   

4.  

   (%i8) trigfactor(-sin(5*x)-cos(3*y));
                           3 y   5 x   %pi      3 y   5 x   %pi
   (%o8)           - 2 cos(--- - --- + ---) cos(--- + --- - ---)
                            2     2     4        2     2     4
   (%i9) trigrat(%);
   (%o9)                      (- cos(3 y)) - sin(5 x)
   

5.  

   (%i10) sin(alpha)+sin(beta)=trigfactor(sin(alpha)+sin(beta));
                                          beta   alpha      beta   alpha
   (%o10)  sin(beta) + sin(alpha) = 2 cos(---- - -----) sin(---- + -----)
                                           2       2         2       2
   

   (%i11) trigrat(%);
   (%o78)          sin(beta) + sin(alpha) = sin(beta) + sin(alpha)
   

6.  

   (%i12) sin(alpha)-sin(beta)=trigfactor(sin(alpha)-sin(beta));
                                           beta   alpha      beta   alpha
   (%o12) sin(alpha) - sin(beta) = - 2 sin(---- - -----) cos(---- + -----)
                                            2       2         2       2
   

7.  

   (%i13) cos(alpha)+cos(beta)=trigfactor(cos(alpha)+cos(beta));
                                          beta   alpha      beta   alpha
   (%o80)  cos(beta) + cos(alpha) = 2 cos(---- - -----) cos(---- + -----)
                                           2       2         2       2
   

8.  

   (%i14) cos(alpha)-cos(beta)=trigfactor(cos(alpha)-cos(beta));
                                          beta   alpha      beta   alpha
   (%o14)  cos(alpha) - cos(beta) = 2 sin(---- - -----) sin(---- + -----)
                                           2       2         2       2
   

9.  

   (%i15) trigfactor(3*sin(x)+7*cos(x));
   (%o15)                        3 sin(x) + 7 cos(x)
   

   (%i16) c2sin(%);
                                                    7
   (%o16)                     sqrt(58) sin(x + atan(-))
                                                    3
   

   (%i17) trigexpand(%),expand;
   (%o17)                        3 sin(x) + 7 cos(x)
   

   10.

   (%i18) trigfactor(sin(2*x));
   (%o18)                             sin(2 x)
   (%i19) trigexpand(%);
   (%o19)                          2 cos(x) sin(x)