Hello, i calculated the value of sin(45) in Matlab, the result was sin(45)=0.8509 and for cosin i got cos(45)= 0.5253. From general mathematics we know that sin(45)=cos(45) then why Matlab is giving different results??
MATLAB: Calculating sin and cos functions
MATLABsine and cosine functions
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qini = zeros(1,6); % Initial guess
ci = fsolve(@funec,qini) % fsolve with the function handle, unknown variables and initial guess
% done many variations here, none of them worked
function ec = funec(x);q1 = x(1);q2 = x(2);q3 = x(3);q4 = x(4);q5 = x(5);q6 = x(6);ec =[... sin(q6)*(cos(q1)*cos(q2)*sin(q3)*sin(q4) - cos(q4)*sin(q1) + cos(q1)*cos(q3)*sin(q2)*sin(q4)) - cos(q6)*(cos(q2 + q3)*cos(q1)*sin(q5) + cos(q5)*sin(q1)*sin(q4) + cos(q1)*cos(q2)*cos(q4)*cos(q5)*sin(q3) + cos(q1)*cos(q3)*cos(q4)*cos(q5)*sin(q2)) - 3097002709122497/18014398509481984 cos(q6)*(cos(q1)*cos(q4) + cos(q2)*sin(q1)*sin(q3)*sin(q4) + cos(q3)*sin(q1)*sin(q2)*sin(q4)) + sin(q6)*(cos(q2 + q3)*sin(q1)*sin(q5) - cos(q1)*cos(q5)*sin(q4) + cos(q2)*cos(q4)*cos(q5)*sin(q1)*sin(q3) + cos(q3)*cos(q4)*cos(q5)*sin(q1)*sin(q2)) - 6124274932042799/36028797018963968 - sin(q2 + q3)*cos(q5) - cos(q2 + q3)*cos(q4)*sin(q5) - 139701203459709/140737488355328 70*cos(q1) + 360*cos(q1)*sin(q2) - 65*sin(q1)*sin(q4)*sin(q5) + 380*cos(q1)*cos(q2)*cos(q3) - 380*cos(q1)*sin(q2)*sin(q3) + 65*cos(q1)*cos(q2)*cos(q3)*cos(q5) - 65*cos(q1)*cos(q5)*sin(q2)*sin(q3) - 65*cos(q1)*cos(q2)*cos(q4)*sin(q3)*sin(q5) - 65*cos(q1)*cos(q3)*cos(q4)*sin(q2)*sin(q5) + 19869/50 sin(q6)*(cos(q1)*cos(q4) + cos(q2)*sin(q1)*sin(q3)*sin(q4) + cos(q3)*sin(q1)*sin(q2)*sin(q4)) - cos(q6)*(cos(q2 + q3)*sin(q1)*sin(q5) - cos(q1)*cos(q5)*sin(q4) + cos(q2)*cos(q4)*cos(q5)*sin(q1)*sin(q3) + cos(q3)*cos(q4)*cos(q5)*sin(q1)*sin(q2)) + 2202239090516653/2251799813685248 cos(q2 + q3)*cos(q6)*sin(q4) - sin(q6)*(sin(q2 + q3)*sin(q5) - cos(q2 + q3)*cos(q4)*cos(q5)) + 7591943296130745/288230376151711744 cos(q2 + q3)*cos(q1)*cos(q5) - sin(q5)*(sin(q1)*sin(q4) + cos(q1)*cos(q2)*cos(q4)*sin(q3) + cos(q1)*cos(q3)*cos(q4)*sin(q2)) - 102014544375725/18014398509481984 70*sin(q1) + 360*sin(q1)*sin(q2) - 380*sin(q1)*sin(q2)*sin(q3) + 380*cos(q2)*cos(q3)*sin(q1) + 65*cos(q1)*sin(q4)*sin(q5) + 65*cos(q2)*cos(q3)*cos(q5)*sin(q1) - 65*cos(q5)*sin(q1)*sin(q2)*sin(q3) - 65*cos(q2)*cos(q4)*sin(q1)*sin(q3)*sin(q5) - 65*cos(q3)*cos(q4)*sin(q1)*sin(q2)*sin(q5) - 1563/20 cos(q6)*(sin(q2 + q3)*sin(q5) - cos(q2 + q3)*cos(q4)*cos(q5)) + cos(q2 + q3)*sin(q4)*sin(q6) - 2129854249224409/18014398509481984 cos(q6)*(cos(q1)*cos(q2)*sin(q3)*sin(q4) - cos(q4)*sin(q1) + cos(q1)*cos(q3)*sin(q2)*sin(q4)) + sin(q6)*(cos(q2 + q3)*cos(q1)*sin(q5) + cos(q5)*sin(q1)*sin(q4) + cos(q1)*cos(q2)*cos(q4)*cos(q5)*sin(q3) + cos(q1)*cos(q3)*cos(q4)*cos(q5)*sin(q2)) - 8872946535830847/9007199254740992 cos(q2 + q3)*cos(q5)*sin(q1) - sin(q5)*(cos(q2)*cos(q4)*sin(q1)*sin(q3) - cos(q1)*sin(q4) + cos(q3)*cos(q4)*sin(q1)*sin(q2)) - 8718732974408059/72057594037927936 360*cos(q2) - 380*cos(q2)*sin(q3) - 380*cos(q3)*sin(q2) - 65*cos(q2)*cos(q5)*sin(q3) - 65*cos(q3)*cos(q5)*sin(q2) - 65*cos(q2)*cos(q3)*cos(q4)*sin(q5) + 65*cos(q4)*sin(q2)*sin(q3)*sin(q5) - 39571/100];endI needed to parameterise your V equation for it to work with the Optimization Toolbox routines:
% % % p(1) = a, p(2) = b;
V = @(p) -(26*(2000*cos((pi*p(1))/180) + (sin((pi*(p(1) + 45))/180)*((2600000*cos((pi*(p(1) + p(2)))/180)*(2^(1/2)/2 + (cos((pi*(p(1) + p(2)))/180)*sin((pi*(p(1) + 45))/180))/sin((pi*p(2))/180)))/(cos((pi*p(1))/180)^2*(cos((pi*(p(1) + p(2)))/180)*sin((pi*p(1))/180) - sin((pi*(p(1) + p(2)))/180)*cos((pi*p(1))/180))) + (2600000*cos((pi*p(1))/180)*sin((pi*(p(1) + 45))/180))/(cos((pi*(p(1) + p(2)))/180)*sin((pi*p(2))/180)*(cos((pi*(p(1) + p(2)))/180)*sin((pi*p(1))/180) - sin((pi*(p(1) + p(2)))/180)*cos((pi*p(1))/180)))))/((1300*(2^(1/2)/2 + (cos((pi*(p(1) + p(2)))/180)*sin((pi*(p(1) + 45))/180))/sin((pi*p(2))/180))^2)/cos((pi*p(1))/180)^3 + (1300*sin((pi*(p(1) + 45))/180)^2)/(cos((pi*(p(1) + p(2)))/180)*sin((pi*p(2))/180)^2) + 8085710131136235/4398046511104)))/(3*cos((pi*(p(1) + p(2)))/180)*sin((pi*p(2))/180)) - (26*((2^(1/2)*((2600000*cos((pi*(p(1) + p(2)))/180)*(2^(1/2)/2 + (cos((pi*(p(1) + p(2)))/180)*sin((pi*(p(1) + 45))/180))/sin((pi*p(2))/180)))/(cos((pi*p(1))/180)^2*(cos((pi*(p(1) + p(2)))/180)*sin((pi*p(1))/180) - sin((pi*(p(1) + p(2)))/180)*cos((pi*p(1))/180))) + (2600000*cos((pi*p(1))/180)*sin((pi*(p(1) + 45))/180))/(cos((pi*(p(1) + p(2)))/180)*sin((pi*p(2))/180)*(cos((pi*(p(1) + p(2)))/180)*sin((pi*p(1))/180) - sin((pi*(p(1) + p(2)))/180)*cos((pi*p(1))/180)))))/(2*((1300*(2^(1/2)/2 + (cos((pi*(p(1) + p(2)))/180)*sin((pi*(p(1) + 45))/180))/sin((pi*p(2))/180))^2)/cos((pi*p(1))/180)^3 + (1300*sin((pi*(p(1) + 45))/180)^2)/(cos((pi*(p(1) + p(2)))/180)*sin((pi*p(2))/180)^2) + 8085710131136235/4398046511104)) + (cos((pi*(p(1) + p(2)))/180)*(2000*cos((pi*p(1))/180) + (sin((pi*(p(1) + 45))/180)*((2600000*cos((pi*(p(1) + p(2)))/180)*(2^(1/2)/2 + (cos((pi*(p(1) + p(2)))/180)*sin((pi*(p(1) + 45))/180))/sin((pi*p(2))/180)))/(cos((pi*p(1))/180)^2*(cos((pi*(p(1) + p(2)))/180)*sin((pi*p(1))/180) - sin((pi*(p(1) + p(2)))/180)*cos((pi*p(1))/180))) + (2600000*cos((pi*p(1))/180)*sin((pi*(p(1) + 45))/180))/(cos((pi*(p(1) + p(2)))/180)*sin((pi*p(2))/180)*(cos((pi*(p(1) + p(2)))/180)*sin((pi*p(1))/180) - sin((pi*(p(1) + p(2)))/180)*cos((pi*p(1))/180)))))/((1300*(2^(1/2)/2 + (cos((pi*(p(1) + p(2)))/180)*sin((pi*(p(1) + 45))/180))/sin((pi*p(2))/180))^2)/cos((pi*p(1))/180)^3 + (1300*sin((pi*(p(1) + 45))/180)^2)/(cos((pi*(p(1) + p(2)))/180)*sin((pi*p(2))/180)^2) + 8085710131136235/4398046511104)))/sin((pi*p(2))/180)))/(3*cos((pi*p(1))/180)^2) - (539047342075749*((2600000*cos((pi*(p(1) + p(2)))/180)*(2^(1/2)/2 + (cos((pi*(p(1) + p(2)))/180)*sin((pi*(p(1) + 45))/180))/sin((pi*p(2))/180)))/(cos((pi*p(1))/180)^2*(cos((pi*(p(1) + p(2)))/180)*sin((pi*p(1))/180) - sin((pi*(p(1) + p(2)))/180)*cos((pi*p(1))/180))) + (2600000*cos((pi*p(1))/180)*sin((pi*(p(1) + 45))/180))/(cos((pi*(p(1) + p(2)))/180)*sin((pi*p(2))/180)*(cos((pi*(p(1) + p(2)))/180)*sin((pi*p(1))/180) - sin((pi*(p(1) + p(2)))/180)*cos((pi*p(1))/180)))))/(43980465111040*((1300*(2^(1/2)/2 + (cos((pi*(p(1) + p(2)))/180)*sin((pi*(p(1) + 45))/180))/sin((pi*p(2))/180))^2)/cos((pi*p(1))/180)^3 + (1300*sin((pi*(p(1) + 45))/180)^2)/(cos((pi*(p(1) + p(2)))/180)*sin((pi*p(2))/180)^2) + 8085710131136235/4398046511104));Since you want to optimise it and the Optimization Toolbox functions minimise it, negating it produced a maximum. (That explains the minus sign.)
The code for the optimisation is:
A = [1 1]; % Constraints
b = 90; % Constraints[angs, fval] = fmincon(V, [1 1]', A, b)asum = angs(1) + angs(2) % Check
The A and b arguments are the equivalent of: (a + b) <= 90. The asum statement simply checks to see that the constraint was met.
The optimisation results are:
angs = -93.3393e+000 180.0000e+000fval = -694.1977e+009asum = 86.6607e+000I negated V to get the maximum, so in reality fval = 694.1977e+009.
Since the solver uses the Jacobian to find the minimum of the function and reported:
Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe default value of the step size tolerance and constraints are satisfied to within the default value of the constraint tolerance.the constraints on the derivatives were satisfied automatically.
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