96.2.5 Solve Trignometric Equations

Function: trigsolve (x) ¶

The function trigsolve find solutions of trigonometric equation from interval \([a,b)\) .

Examples:

1.  

   (%i38) eq:eq:3*sin(x)+4*cos(x)=2;
   (%o38)                      3 sin(x) + 4 cos(x) = 2
   
   (%i39) plot2d([3*sin(x)+4*cos(x),2],[x,-%pi,%pi]);
   
   

   

   
   (%o39)
   (%i40) sol:trigsolve(eq,-%pi,%pi);
                     2 sqrt(21)   12              2 sqrt(21)   12
   (%o40)      {atan(---------- - --), %pi - atan(---------- + --)}
                         5        5                   5        5
   (%i41) float(%), numer;
   (%o41)            {- 0.5157783719341241, 1.802780589520693}
   

   Answ. : \(x = \tan^{-1}\left({2\sqrt{21}\over 5} - {12\over 5}\right) + 2\pi k\) ; \(x = \pi - \tan^{-1}\left({2\sqrt{21}\over 5} + {12\over 5}\right) + 2\pi k,\) \(k\) – any integer.

2.  

   (%i6) eq:cos(3*x)-sin(x)=sqrt(3)*(cos(x)-sin(3*x));
   (%o6)         cos(3 x) - sin(x) = sqrt(3) (cos(x) - sin(3 x))
   (%i7) plot2d([lhs(eq)-rhs(eq)], [x,0,2*%pi])$
   
   

   

   We have 6 solutions from [0, 2*pi].

   (%i8) plot2d([lhs(eq)-rhs(eq)], [x,0.2,0.5]);
   
   

   

   
   (%i9) plot2d([lhs(eq)-rhs(eq)], [x,3.3,3.6]);
   
   

   

   
   (%i10) trigfactor(lhs(eq))=map(trigfactor,rhs(eq));
                      %pi            %pi                      %pi            %pi
   (%o15) - 2 sin(x + ---) sin(2 x - ---) = 2 sqrt(3) sin(x - ---) sin(2 x - ---)
                       4              4                        4              4
   (%i11) factor(lhs(%)-rhs(%));
                    4 x + %pi                4 x - %pi       8 x - %pi
   (%o11)  - 2 (sin(---------) + sqrt(3) sin(---------)) sin(---------)
                        4                        4               4
   

   Equation is equivalent to

   (%i12) L:factor(rhs(%)-lhs(%));
                   4 x + %pi                4 x - %pi       8 x - %pi
   (%o12)   2 (sin(---------) + sqrt(3) sin(---------)) sin(---------)
                       4                        4               4
   

   (%i13) eq1:part(L,2)=0;
                        4 x + %pi                4 x - %pi
   (%o13)           sin(---------) + sqrt(3) sin(---------) = 0
                            4                        4
   

   (%i14) eq2:part(L,3)=0;
                                    8 x - %pi
   (%o14)                       sin(---------) = 0
                                        4
   

   (%i15) S1:trigsolve(eq1,0,2*%pi);
                                    %pi  13 %pi
   (%o15)                         {---, ------}
                                    12     12
   (%i16) S2:trigsolve(eq2,0,2*%pi);
                              %pi  5 %pi  9 %pi  13 %pi
   (%o16)                   {---, -----, -----, ------}
                               8     8      8      8
   (%i17) S:listify(union(S1,S2));
                      %pi  %pi  5 %pi  13 %pi  9 %pi  13 %pi
   (%o17)            [---, ---, -----, ------, -----, ------]
                      12    8     8      12      8      8
   (%i18) float(%), numer;
   (%o18) [0.2617993877991494, 0.3926990816987241, 1.963495408493621, 
                         3.403392041388942, 3.534291735288517, 5.105088062083414]
   

   Answer: \(x = a + 2\pi k,\) where \(a\) any from \(S\), \(k\) any integer.

3.  

   (%i19) eq:8*cos(x)*cos(4*x)*cos(5*x)-1=0;
   (%o19)               8 cos(x) cos(4 x) cos(5 x) - 1 = 0
   
   (%i20) trigrat(%);
   (%o20)          2 cos(10 x) + 2 cos(8 x) + 2 cos(2 x) + 1 = 0
   

   Left side is periodic with period \(T=\pi\) .

   We have 10 solutions from [0, pi].

   (%i21) plot2d([lhs(eq),rhs(eq)],[x,0,%pi]);
   
   

   

   
   (%i22) x4:find_root(eq, x, 1.3, 1.32);
   (%o22)                        1.308996938995747
   (%i23) x5:find_root(eq, x, 1.32, 1.35);
   (%o23)                        1.346396851538483
   (%i24) plot2d([lhs(eq),0], [x,1.3,1.35], [gnuplot_preamble, "set grid;"]);
   
   

   

   Equation we multiply by \(2\sin x\cos 2x\) :

   (%i25) eq*2*sin(x)*cos(2*x);
   (%o25)     2 sin(x) cos(2 x) (8 cos(x) cos(4 x) cos(5 x) - 1) = 0
   (%i26) eq1:trigreduce(%),expand;
   (%o26)                     sin(13 x) + sin(x) = 0
   

   (%i27) trigfactor(lhs(eq1))=0;
   (%o27)                     2 cos(6 x) sin(7 x) = 0
   

   (%i28) S1:trigsolve(cos(6*x),0,%pi);
                       %pi  %pi  5 %pi  7 %pi  3 %pi  11 %pi
   (%o28)             {---, ---, -----, -----, -----, ------}
                       12    4    12     12      4      12
   

   (%i29) S2:trigsolve(sin(7*x),0,%pi);
                        %pi  2 %pi  3 %pi  4 %pi  5 %pi  6 %pi
   (%o29)           {0, ---, -----, -----, -----, -----, -----}
                         7     7      7      7      7      7
   

   We remove solutions of \(\sin x = 0\) and \(\cos 2x = 0.\)

   (%i30) S3:trigsolve(sin(x),0,%pi);
   (%o30)                               {0}
   (%i31) S4:trigsolve(cos(2*x),0,%pi);
                                    %pi  3 %pi
   (%o31)                          {---, -----}
                                     4     4
   

   We find 10 solutions from \([0, \pi]\) :

   (%i32) union(S1,S2)$ setdifference(%,S3)$ setdifference(%,S4);
            %pi  %pi  2 %pi  5 %pi  3 %pi  4 %pi  7 %pi  5 %pi  6 %pi  11 %pi
   (%o34) {---, ---, -----, -----, -----, -----, -----, -----, -----, ------}
            12    7     7     12      7      7     12      7      7      12
   

   (%i35) S:listify(%);
           %pi  %pi  2 %pi  5 %pi  3 %pi  4 %pi  7 %pi  5 %pi  6 %pi  11 %pi
   (%o35) [---, ---, -----, -----, -----, -----, -----, -----, -----, ------]
           12    7     7     12      7      7     12      7      7      12
   

   (%i36) length(S);
   (%o36)                               10
   (%i37) float(S), numer;
   (%o37) [0.2617993877991494, 0.4487989505128276, 0.8975979010256552, 
   1.308996938995747, 1.346396851538483, 1.79519580205131, 1.832595714594046, 
   2.243994752564138, 2.692793703076966, 2.879793265790644]
   

   Answer: \(x = a + 2\pi k,\) where \(a\) any from \(S\), \(k\) any integer.