which solver is best to solve a set of trig equation?

I wonder if existing MATLAB solvers can solve my set of trignometric equations:

For the above equations, assume I know , , , and , and I want to solve for θ and ϕ.

I tried using fsolve with 2 and 3 equations but the solutions I got was incorrect:

%test
lambda = 0.06;
azi = 36;
ele = 55;
%DOA= [azi ele];
x1 = 0.06; y1 = 0 ;  z1 = 0;
x2 = 0.12; y2 = 0; z2 = 0;
x3 = 0.06; y3 = 0.06; z3 = 0;
s1 =  exp((2*pi*1j)*((x1*cosd(azi)*sind(ele)+y1*sind(azi)*sind(ele)+z1*cosd(ele))/lambda))

s1 = -0.5214 - 0.8533i

s2 =  exp((2*pi*1j)*((x2*cosd(azi)*sind(ele)+y2*sind(azi)*sind(ele)+z2*cosd(ele))/lambda))

s2 = -0.4563 + 0.8898i

s3 =  exp((2*pi*1j)*((x3*cosd(azi)*sind(ele)+y3*sind(azi)*sind(ele)+z3*cosd(ele))/lambda))

s3 = 0.6169 + 0.7870i

%now, work backward, use s1 and s2 ro find azi and ele, still try to use
%fsolve
leftside1  = real(log(s1)/(2*pi*1j)) %solution contains imaginary value == 0i 

leftside1 = -0.3373

leftside2  = real(log(s2)/(2*pi*1j))

leftside2 = 0.3254

leftside3  = real(log(s3)/(2*pi*1j))

leftside3 = 0.1442

((x1*cosd(azi)*sind(ele)+y1*sind(azi)*sind(ele)+z1*cosd(ele))/lambda)-leftside1

ans = 1

((x2*cosd(azi)*sind(ele)+y2*sind(azi)*sind(ele)+z2*cosd(ele))/lambda)-leftside2

ans = 1

((x3*cosd(azi)*sind(ele)+y3*sind(azi)*sind(ele)+z3*cosd(ele))/lambda)-leftside3

ans = 1

options = optimoptions('fsolve','Display','none','PlotFcn',@optimplotfirstorderopt,'Algorithm','levenberg-marquardt')

options = 
  fsolve options:

   Options used by current Algorithm ('levenberg-marquardt'):
   (Other available algorithms: 'trust-region', 'trust-region-dogleg')

   Set properties:
                   Algorithm: 'levenberg-marquardt'
                     Display: 'none'
                     PlotFcn: @optimplotfirstorderopt

   Default properties:
              CheckGradients: 0
    FiniteDifferenceStepSize: 'sqrt(eps)'
        FiniteDifferenceType: 'forward'
           FunctionTolerance: 1.0000e-06
      MaxFunctionEvaluations: '200*numberOfVariables'
               MaxIterations: 400
                   OutputFcn: []
    SpecifyObjectiveGradient: 0
               StepTolerance: 1.0000e-06
                    TypicalX: 'ones(numberOfVariables,1)'
                 UseParallel: 0

   Show options not used by current Algorithm ('levenberg-marquardt')

fun = @(DOA)f_angle(DOA,leftside1,leftside2,leftside3);
DOA0 = [35,54];
DOA = fsolve(fun,DOA0,options)

Error using uicontrol
This functionality is not available on remote platforms.

Error in callAllOptimPlotFcns (line 78)
stopBtn = uicontrol('string',getString(message('MATLAB:optimfun:funfun:optimplots:ButtonStop')), ...

Error in levenbergMarquardt>callOutputAndPlotFcns (line 437)
stop = callAllOptimPlotFcns(plotfcns,xOutputfcn,optimValues,state,varargin{:}) || stop;

Error in levenbergMarquardt (line 147)
[optimValues, stop] = callOutputAndPlotFcns(outputfcn,plotfcns,caller,reshape(XOUT,xShape),...

Error in fsolve (line 450)
levenbergMarquardt(funfcn,x,lb,ub,verbosity,options,defaultopt,f,JAC,caller, ...

function f =  f_angle(DOA,leftside1,leftside2,leftside3)
    lambda = 0.06;
    x1 = 0.06; y1 = 0 ;  z1 = 0;
    x2 = 0.12; y2 = 0; z2 = 0;
    x3 = 0.06; y3 = 0.06; z3 = 0;
    f(1)= ((x1*cosd(DOA(1))*sind(DOA(2))+y1*sind(DOA(1))*sind(DOA(2))+z1*cosd(DOA(2)))/lambda) -leftside1;
    f(2)= ((x2*cosd(DOA(1))*sind(DOA(2))+y2*sind(DOA(1))*sind(DOA(2))+z2*cosd(DOA(2)))/lambda) -leftside2;
    f(3)= ((x3*cosd(DOA(1))*sind(DOA(2))+y3*sind(DOA(1))*sind(DOA(2))+z3*cosd(DOA(2)))/lambda) -leftside3;
end

my intended angle is [36 55] but fsolve returns [52.4154 5.9013].

Ultimately, my value would contain some small noise so equal sign would turn into an approx equal sign so I think symbolic solver would be in no use.

Would be nice to know whether this set of equations are solvable using MATLAB? If so which solver/set up should I be looking into?

Thanks

[Torsten]

As you can see above, f(1) = f(2) = f(3) = 1, not 0.

[Xingda Chen]

Hi Torsten,

Yes I realized I cannot simiply log a complex number. I am editing my script such that my a+bj complex number returns a phase reading in term of degrees/radians, and then re-run fsolve and see if that works

Answer 1

[Xingda Chen]

I realized I cannot simiply log a complex number. I am editing my script such that my a+bj complex number returns an angle reading in term of degrees/radians, as well as making sure the returned phase isn't wrap within 2*pi. fsolve() proudly completed the job.... kudos to the team who developed fsolve()!