How to solve complex equations?

I am trying to solve the following equation.

I have the following script to solve for theta, but getting erro when I am running the program.

Any suggestions is much appreciated.

Thank you.

clc;
clear;
close all;


syms theta 
assume(0 <= theta <= 2*pi)
t = 0.4e-3;
n = 1.523;
delta_y = 3;

S1 = solve(delta_y == t*sind(theta)*(1 - sqrt((1-sind(theta)^2)/n^2 - sind(theta)^2)),theta);

[John D'Errico]

You don't say what that equation should model. While the picture seems to show the variables, it shows what appears to be a rectangle canted at an angle. But one edge of the rectangle seems to have length t, and the other parallel edge shows that as n. So your drawing does not make sense. Perhaps n and t do not mean what they are shown as. I might guess that n refers to one dimension of the rect, and t is the other. But that is just a wild guess. But then one cannot possibly derive the equation, or show that it is correct.

[Torsten]

You forgot a bracket around the denominator under the square root.

[Navaneeth]

Sorry that I left out some details. N here is refractive index of the material and t is the thickness of the slab...I want to find the theta in angles for a fixed shift of delta_y.

Thank you.

Answer 1

[Torsten]

Your equation has only complex solutions.

Note that you used sind instead of sin in the function definition. Thus assuming theta in radians is wrong.

clc;
clear;
close all;


syms theta 
%assume(0 <= theta <= 2*pi)
t = 0.4e-3;
n = 1.523;
delta_y = 3;

S1 = solve(delta_y == t*sind(theta)*(1 - sqrt((1-sind(theta)^2)/(n^2 - sind(theta)^2))),theta,'ReturnConditions',1,'MaxDegree',3)

S1 = struct with fields:
         theta: [2×1 sym]
    parameters: k
    conditions: [2×1 sym]

vpa(S1.theta)

ans = 

S1.parameters

ans = 

k

S1.conditions

ans = 

[Navaneeth]

Thank you, excuse me for asking what are these parameters and condition and how can I get the theta value in simple degrees from this format.

[Torsten]

The solutions in "degrees" are

360*k + 270 - 56.305582...*1i
-360*k - 90 + 56.305582...*1i

where 1i is the imaginary unit and k is any integer.

As said, a real solution does not exist in the case of your parameters.

[Navaneeth]

Thank you, but solving for delta_y for a fixed theta value gives me a real value

clc;
clear;
close all;

syms delta_y_raw
t = 0.7e-3; %thickness
theta = 5; %angle  
n = 1.533; %refractive index of the material.
d_y  = solve(delta_y_raw == t*sind(theta)*(1 - sqrt((1-sind(theta)^2)/(n^2 - sind(theta)^2))), delta_y_raw);
delta_y = double(d_y)/1e-6 % in um

delta_y = 21.2991

Now to find theta for different delta_y how to use this equation in such a way that I dont get complex theta solutions.

Thank you.

[Torsten]

If you use the new parameters, you get back some real solutions:

clc;
clear;
close all;


syms theta 
t = 0.7e-3;
n = 1.533;
delta_y = 21.2991*1e-6;

S1 = vpa(solve(delta_y == t*sind(theta)*(1 - sqrt((1-sind(theta)^2)/(n^2 - sind(theta)^2))),theta,'MaxDegree',3))

S1 = 

format long
S1 = double(S1(abs(imag(S1))<1e-6))        

S1 = 
   1.0e+02 *

  1.749999973612585 - 0.000000000000000i
  0.050000026387415 + 0.000000000000000i

[Navaneeth]

Thank you, this is exactly what I wanted.