How do I solve this set of equations using MATLAB?

I am trying to solve the following equations for the variable :

and

for that I wrote the following:

syms M n_1 n_2 n_3 th1 th2 th3

eqns = [M == ( (n_2*cos(th2) - n_3*cos(th3))/(n_2*cos(th2) + n_3*cos(th3)) )*...
( (n_2*cos(th2) + n_1*cos(th1))/(n_2*cos(th2) - n_1*cos(th1)) ) , ...
n_1*sin(th1) == n_2*sin(th2) ,  n_1*sin(th1) == n_3*sin(th3) ];

S = solve(eqns, [n_2]) 

But when I run the code I get that:

Empty sym: 0-by-1

What have I done wrong? How do I solve multiple equations?

Answer 1

[Torsten]

You want to solve for one unknown - thus you only need one of the two equations.

If you want to solve for n1, n2 and n3, you have to specify this is the solve command.

But MATLAB cannot find an analytical solution, as you can see below.

syms M n_1 n_2 n_3 th1 th2 th3
eqns = [M == ( (n_2*cos(th2) - n_3*cos(th3))/(n_2*cos(th2) + n_3*cos(th3)) )*...
        ( (n_2*cos(th2) + n_1*cos(th1))/(n_2*cos(th2) - n_1*cos(th1)) ) , ...
        n_1*sin(th1) == n_2*sin(th2) ,  n_1*sin(th1) == n_3*sin(th3) ];
S = solve(eqns, [n_1,n_2,n_3]) 

S = struct with fields:
    n_1: [0×1 sym]
    n_2: [0×1 sym]
    n_3: [0×1 sym]

[Goncalo Costa]

Thank you so much for your help. Could this be a limitation of MATLAB, as there is definitely a solution? Or could it be that I should rewrite the equations in some other way?

[Torsten]

Where do the equations stem from ? Maybe the background could give an idea for its solution.

Multiplying by the denominator in equation 1, MATLAB only computes n_1 = n_2 = n_3 = 0 as solution.

[Goncalo Costa]

These are from the optical properties of a triple-layer in a reflection geometry. Previous equations, simiar to this one have been solved, and physically there ought to be a solution, I just can't solve it by hand. I tried Mathematica but the program keeps crashing, and now I tried MATLAB to no avail.

[John D'Errico]

I would note there is pretty much never an analytical solution to a problem, where a variable appears both inside and outside of a trig function. Thus anything of the general form x*sin(x) = c, will generally have no algebraic/analytical solution, even if it has a numerical solution.

[Torsten]

And this is the case somewhere ? The n_i are all outside the trig expressions.

Answer 2

[Torsten]

As you can see from the below analysis, there is only the solution n_1 = n_2 = n_3 = 0 without conditions on the angles th1, th2 and th3.

syms M n_1 n_2 n_3 th1 th2 th3
f = M *(n_2*cos(th2) + n_3*cos(th3))*(n_2*cos(th2) - n_1*cos(th1)) - (n_2*cos(th2) - n_3*cos(th3))*(n_2*cos(th2) + n_1*cos(th1));
f = subs(f,[n_2 n_3],[n_1*sin(th1)/sin(th2),n_1*sin(th1)/sin(th3)])

f = 

f = simplify(f)

f =