Find a variable as function of other variable with equations system using Symbolic ToolBox

I have 3 equations that connects between 4 variables, let's say:

syms a b c d;

r=b+d;

e=pi/2;

eq1 = c - a*sin(d) + b*cos(d) == 0;
eq2 = r - c*tan(e) - a*cos(d) - b*sin(d) == 0;
eq3 = b == e*r + a*tan(e);

Now I would like to find c(d), means, I would like to find what is c as a function of other variable, in this case - d.

Then perform other operations on c(d) such as differntiate etc.

What would be the best/easiest way to do that using Symbolic Toolbox?

Thanks.

[Walter Roberson]

10 radians is an unusual angle, but that is probably not so important for your question.

[Evyatar Sivan]

edited

Answer 1

[Walter Roberson]

syms a b c d;

r = b+d;

e = sym(pi)/10;   %not pi/2, that does not have a solution!

eq1 = c - a*sin(d) + b*cos(d) == 0;
eq2 = r - c*tan(e) - a*cos(d) - b*sin(d) == 0;
eq3 = b == e*r + a*tan(e);

sol = solve([eq1,eq2,eq3], [a, b, c], 'returnconditions', true);
c = symfun(simplify(sol.c), d);

This will be valid except at four sets of d values whose values can be deduced by analyzing sol.conditions (in each case adding an integer multiple of 2*pi gets another forbidden value)

[Evyatar Sivan]

Thank you so much Walter.

looks like it's working good, but still for some reason I get complex result which I know is not the answer.

I've been over this for hours... wish you might be able to help if there's anything wrong with the code.

my code:

clc;
clear all;

% Unknown Variables

syms Pa N T theta;

% Given Variables

L=5.07/sin(theta);

delta=deg2rad(10);

c=10;

gamma=20;

phi=deg2rad(20);

Q=50.7/tan(theta);

W=((5.07/tan(theta))*0.93+5.07*(5.07/tan(theta))*0.5)*gamma;

% Equations

eq1 = Pa-N*sin(theta)+T*cos(theta) == 0;

eq2 = Q+W-Pa*tan(delta)-N*cos(theta)-T*sin(theta) == 0;

eq3 = T == c*L + N*tan(phi);

% Solution

sol = solve([eq1,eq2,eq3], [N, T, Pa], 'returnconditions', true);
Pa_theta = symfun(simplify(sol.Pa), theta);

eq4 = diff(Pa_theta,theta) == 0;

theta_max = solve(eq4,theta); % gives 4 possible solutions

rad2deg(theta_max(1)) % not quite what I expected... suppose to be 53 degress.

Thank you a lot.

[Walter Roberson]

double(rad2deg(theta_max(3))
ans =

           53.0352986075461 -   6.4856523980211e-71i

The complex portion is due to round-off error.

However, I recommend that you rewrite your code to cut down on the repeated numeric constants, and that you switch into symbolic mode for greater accuracy. Using solve() with equations with floating point constants is always a category mistake: solve() is for exact solutions, and floating point constants are by definition only approximations (except for the ones that are integers.)

Z = @(v) sym(v);

% Unknown Variables

syms Pa N T theta;

% Given Variables

factor507 = Z(5.07);
factor93 = Z(0.93);
L = factor507/sin(theta);

delta = deg2rad(Z(10));

c = Z(10);

gamma = Z(20);

phi = deg2rad(Z(20));

Q = c*factor507/tan(theta);

W = ((factor507/tan(theta))*factor93 + factor507*(factor507/tan(theta))*Z(0.5))*gamma;

% Equations

eq1 = Pa-N*sin(theta)+T*cos(theta) == 0;

eq2 = Q+W-Pa*tan(delta)-N*cos(theta)-T*sin(theta) == 0;

eq3 = T == c*L + N*tan(phi);

% Solution

sol = solve([eq1,eq2,eq3], [N, T, Pa], 'returnconditions', true);
Pa_theta = symfun(simplify(sol.Pa), theta);

eq4 = diff(Pa_theta,theta) == 0;

theta_max = solve(eq4,theta);
rad2deg(theta_max)

Answer 2

[Ameer Hamza]

try this example

syms a b c d;
r=b+d;
e=pi/2;
eq1 = c - a*sin(d) + b*cos(d) == 0;
eq2 = r - c*tan(e) - a*cos(d) - b*sin(d) == 0;
eq3 = b == e*r + a*tan(e);

sol = solve([eq1 eq2 eq3], [a b c]);

C = sol.c; % c as function of d
% Now find its derivative
dC_dd = diff(C,d); % dc/dd
% you can also integrate it
IC = int(C,d);