How do I solve this set of equations using MATLAB?

Question

Goncalo Costa 2022-11-8

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https://ww2.mathworks.cn/matlabcentral/answers/1846168-how-do-i-solve-this-set-of-equations-using-matlab

评论： Torsten 2022-11-8

在 MATLAB Online 中打开

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?

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Answer 1

Torsten 2022-11-8

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https://ww2.mathworks.cn/matlabcentral/answers/1846168-how-do-i-solve-this-set-of-equations-using-matlab#answer_1094113

在 MATLAB Online 中打开

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]

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John D'Errico 2022-11-8

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.