How can I solve three non-linear equations having 3 unknowns in MATLAB?

My non-linear equations are---

cos(a)-cos(b)+cos(c)=0.725
cos(5a)-cos(5b)+cos(5c)=0.5
cos(7a)-cos(7b)+cos(7c)=0.5

[Walter Roberson]

You can see immediately that the solutions will need to be complex, since the sum or difference of three cos() of rational values must be in the range 3*(-1..+1) but the result is required to be -27.27

[Rishikesh Datar]

The correction is made in the question. It is 0.725 instead of -27.27. But is it having unique solution or infinite solutions?

Answer 1

[Star Strider]

Use the Optimization Toolbox fsolve function:

% MAPPING: x(1) = a,  x(2) = b,  x(3) = c
fcn = @(x) [cos(x(1))-cos(x(2))+cos(x(3)) + 27.27; cos(5*x(1))-cos(5*x(2))+cos(5*x(3)) - 0.5; cos(7*x(1))-cos(7*x(2))+cos(7*x(3)) - 0.5];
x0 = [1; 1; 1];
[x, fv] = fsolve(fcn, x0);

Be sure to see the documentation for optimoptions and specifically MaxFunEvals, since fsolve encountered that limit before it was happy with the solution (assuming one exists).

[Walter Roberson]

There are 18 sets of solutions, and in every solution all of the values are complex -- not a single real-valued variable anywhere in any of the solutions.

Answer 2

[Walter Roberson]

For the revised question, there are 18 real-valued solutions, none of which have a closed form solution, as they involve the roots of a polynomial of degree 9. You might be able to find them using the Symbolic Toolbox and solve(), followed with double()

The solutions are approximately

a = 1.347290041624088, b = 1.487303536560025, c = .9437620688704266
a = .9437620688704266, b = 1.487303536560025, c = 1.347290041624088
a = 1.460079617010332, b = 1.179429186973846, c = 0.8989807405828178e-1
a = 0.8989807405828178e-1, b = 1.179429186973846, c = 1.460079617010332
a = .9096157489766238, b = .6148549454499392, c = .3823026965271043
a = .3823026965271043, b = .6148549454499392, c = .9096157489766238
a = 0.8989807405828178e-1, b = 1.681513036579461, c = 1.962163466615947
a = 1.962163466615947, b = 1.681513036579461, c = 0.8989807405828178e-1
a = .9437620688704266, b = 1.794302611965706, c = 1.654289117029769
a = 1.654289117029769, b = 1.794302611965706, c = .9437620688704266
a = 1.347290041624088, b = 2.197830584719367, c = 1.654289117029769
a = 1.654289117029769, b = 2.197830584719367, c = 1.347290041624088
a = .3823026965271043, b = 2.231976904613169, c = 2.526737708139854
a = 2.526737708139854, b = 2.231976904613169, c = .3823026965271043
a = .9096157489766238, b = 2.759289957062689, c = 2.526737708139854
a = 2.526737708139854, b = 2.759289957062689, c = .9096157489766238
a = 1.460079617010332, b = 3.051694579531511, c = 1.962163466615947
a = 1.962163466615947, b = 3.051694579531511, c = 1.460079617010332

[Rishikesh Datar]

Are the values in radians or degree? The values of a, b and c are angles of firing for particular device so they should lie form 0 to 90 degree maximum and good initial guess is required.