MATLAB: Solving difficult trigonmetric equations for theta.

MATLABtrigonometry solutionsvertical tangent lines

First, thank you for taking the time to read my problem. I apprecaiate any help.

Trying to find roots of the below equation. It doesn't seem to work on octave. When I use an online tool like wolframalpha, I can get the asnwer that I am looking for. I can always approximate the answer graphically but I am interested in the code to cruch answers directly. The below is the input and output. There are roots to my problem. I'm not sure why Matlab can't solve for them. The larger picture problem that I am working on is in finding vertical tangent lines to polar equations which I have done on paper but would like to code.

solve(-1*sin(x)*(4 + 2*cos(3*x + 21*3.14/6)) - cos(x)*6*sin(3*x + 21*3.14/6) == 0,x)

ans = {}(0x0)

Interestingly enough, when I do a simple example like the below, I get the obvious solution. So, I know that "solve" at least works with simple examples:

solve(sin(x) == 0,x)
ans = (sym 2×1 matrix)
  ⎡0⎤
  ⎢ ⎥
  ⎣π⎦

Best Answer

polyxpoly works quite well for this situation. I use it here to find the intersection of two lines; your equation (over a specified range) and y=0

range=[0 100];
x=linspace(range(1),range(2),10000); %generate data in a given range
y=-1*sin(x).*(4 + 2*cos(3*x + 21*3.14/6)) - cos(x).*6.*sin(3*x + 21*3.14/6);
[xroots,y0] = polyxpoly(x,y,range,[0 0]); % find intersection
plot(x,y), hold on, plot(xroots,y0,'or');
xroots

Related Solutions

MATLAB: FSOLVE requires all values returned by functions to be of data type double.

If you want to solve a system of equations symbolically or using symbolic variables, use the solve function instead of fsolve. The fsolve function requires that the output of the function you specify as the first input be double precision numbers, not symbolic expressions.

If you must use fsolve with symbolic variables inside the function, you must convert those symbolic expressions into numbers using double before returning the numbers (not the symbolic expressions) from the function.

As a very simple example, put all this code into a function example373677.m and run it:

function y = example373677
y = fsolve(@mysubfun, 3);
 
function z = mysubfun(q)
x = sym('x');
r = x.^3-2*x-5;
z = double(subs(r, q));

Of course, with solve this is easier.

syms x
y = solve(x.^3-2*x-5)

You may need to use vpa to approximate y numerically.

vpa(y)

One of the solutions returned in y should match the result you received from running the function.

MATLAB: Symbolic Math shows weird behavior – not seen in Wolfram or Octave

It looks like the expected solution can be obtained with some work. Maybe someone else can get there faster.

>> syms x real
>> A = sym([1 -2*cos(x) 0; -sin(x)/cos(x)*sin(x) 1 cos(x); 1 -2*sin(x)/cos(x)*sin(x) 2])
 
A =
 
[                1,            -2*cos(x),      0]
[ -sin(x)^2/cos(x),                    1, cos(x)]
[                1, -(2*sin(x)^2)/cos(x),      2]
>> [V,D]=eig(A)
 
V =
 
[ ((cos(x)^4 + 3*cos(x)^2*sin(x)^2)*(1/(9*(((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)) + (((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3) + 4/3))/(cos(x)^4 - 2*sin(x)^6) - (2*(cos(x)^4 + cos(x)^2*sin(x)^4 + cos(x)^2*sin(x)^2))/(cos(x)^4 - 2*sin(x)^6) - (cos(x)^2*sin(x)^2*(1/(9*(((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)) + (((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3) + 4/3)^2)/(cos(x)^4 - 2*sin(x)^6), - (2*(cos(x)^4 + cos(x)^2*sin(x)^4 + cos(x)^2*sin(x)^2))/(cos(x)^4 - 2*sin(x)^6) - ((cos(x)^4 + 3*cos(x)^2*sin(x)^2)*(1/(18*(((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)) + (((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)/2 + (3^(1/2)*(1/(9*(((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)) - (((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3))*1i)/2 - 4/3))/(cos(x)^4 - 2*sin(x)^6) - (cos(x)^2*sin(x)^2*(1/(18*(((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)) + (((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)/2 + (3^(1/2)*(1/(9*(((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)) - (((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3))*1i)/2 - 4/3)^2)/(cos(x)^4 - 2*sin(x)^6), - (2*(cos(x)^4 + cos(x)^2*sin(x)^4 + cos(x)^2*sin(x)^2))/(cos(x)^4 - 2*sin(x)^6) - ((cos(x)^4 + 3*cos(x)^2*sin(x)^2)*(1/(18*(((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)) + (((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)/2 - (3^(1/2)*(1/(9*(((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)) - (((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3))*1i)/2 - 4/3))/(cos(x)^4 - 2*sin(x)^6) - (cos(x)^2*sin(x)^2*(1/(18*(((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)) + (((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)/2 - (3^(1/2)*(1/(9*(((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)) - (((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3))*1i)/2 - 4/3)^2)/(cos(x)^4 - 2*sin(x)^6)]
[      ((3*cos(x)^3 + 2*cos(x)*sin(x)^4)*(1/(9*(((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)) + (((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3) + 4/3))/(2*(cos(x)^4 - 2*sin(x)^6)) - (cos(x)^3 + 2*cos(x)*sin(x)^4 + cos(x)^3*sin(x)^2)/(cos(x)^4 - 2*sin(x)^6) - (cos(x)^3*(1/(9*(((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)) + (((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3) + 4/3)^2)/(2*(cos(x)^4 - 2*sin(x)^6)),      - (cos(x)^3 + 2*cos(x)*sin(x)^4 + cos(x)^3*sin(x)^2)/(cos(x)^4 - 2*sin(x)^6) - ((3*cos(x)^3 + 2*cos(x)*sin(x)^4)*(1/(18*(((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)) + (((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)/2 + (3^(1/2)*(1/(9*(((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)) - (((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3))*1i)/2 - 4/3))/(2*(cos(x)^4 - 2*sin(x)^6)) - (cos(x)^3*(1/(18*(((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)) + (((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)/2 + (3^(1/2)*(1/(9*(((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)) - (((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3))*1i)/2 - 4/3)^2)/(2*(cos(x)^4 - 2*sin(x)^6)),      - (cos(x)^3 + 2*cos(x)*sin(x)^4 + cos(x)^3*sin(x)^2)/(cos(x)^4 - 2*sin(x)^6) - ((3*cos(x)^3 + 2*cos(x)*sin(x)^4)*(1/(18*(((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)) + (((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)/2 - (3^(1/2)*(1/(9*(((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)) - (((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3))*1i)/2 - 4/3))/(2*(cos(x)^4 - 2*sin(x)^6)) - (cos(x)^3*(1/(18*(((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)) + (((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)/2 - (3^(1/2)*(1/(9*(((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)) - (((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3))*1i)/2 - 4/3)^2)/(2*(cos(x)^4 - 2*sin(x)^6))]
[                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           1,                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         1,                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         1]
 
 
D =
 
[ 1/(9*(((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)) + (((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3) + 4/3,                                                                                                                                                                                                                                                                                                                                                                                           0,                                                                                                                                                                                                                                                                                                                                                                                           0]
[                                                                                                                                                                                     0, 4/3 - (((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)/2 - (3^(1/2)*(1/(9*(((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)) - (((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3))*1i)/2 - 1/(18*(((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)),                                                                                                                                                                                                                                                                                                                                                                                           0]
[                                                                                                                                                                                     0,                                                                                                                                                                                                                                                                                                                                                                                           0, 4/3 - (((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)/2 + (3^(1/2)*(1/(9*(((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)) - (((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3))*1i)/2 - 1/(18*(((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3))]
 
>> D=simplify(diag(D),300)  % unclear why D(1) doesn't simplify to 2 - 1i
 
D =
 
 2 - (2*(-1)^(1/3)*(2*3^(1/2) - 3)*(- 15*3^(1/2) - 26)^(1/3))/3 - 3^(1/2)/3
                                                                          0
                                                                     2 + 1i
 
>> V2=V(:,2);  % pick the eigenvector associated with lamda = 0
>> simplify(imag(V2),100)  % verify the imaginary part is 0
 
ans =
 
 0
 0
 0
 
>> V2=real(V2);  % grab the real part
>> V2=simplify(V2,100)
 
V2 =
 
         - 1/cos(2*x) - 1
 -cos(x)/(2*cos(x)^2 - 1)
                        1
 
 
>> simplify(V2*(1-2*sin(x)^2),100)
 
ans =
 
 -2*cos(x)^2
     -cos(x)
    cos(2*x)

Best Answer

Related Solutions

MATLAB: FSOLVE requires all values returned by functions to be of data type double.

MATLAB: Symbolic Math shows weird behavior – not seen in Wolfram or Octave

Related Question