# Can not get exact exppresion due to cos function using Matlab symbolic toolbox

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K Hu on 26 Oct 2020
Edited: K Hu on 26 Oct 2020
hello,
I am currently using Matlab to inplemente a groupe of function which is showing in the next picture.
briefly, this is a conditional expression with two variable, , repectively.
The objective is generate is expression assuming that σand T are time depending variable then drivate the expression to get the ODE form expression.
here is my code, i found an odd thing that the signal due to the term in expression 'g13(t)' rests always postive, even i'd well defined as a negative term.
syms xi_S0 sigma_crs sigma_crf xi_T0 C_M C_A M_s M_f A_s A_f Delta
syms g11(t) g12(t) g13(t) g14(t) T(t) sigma(t) a_A
%a_A=pi/(A_f-A_s);
a_M=pi/(M_s-M_f);
%Delta= (1-xi_T0-xi_S0)/2 * cos(a_M*(T-M_f)) + (1 + xi_T0 - xi_S0)/2
%case 1
g11(t) =xi_T0/(1-xi_S0) + (1-(xi_T0+xi_S0))/(1-xi_S0) * ((1-xi_S0) /2 * cos(pi/(sigma_crs-sigma_crf) * (sigma(t)-sigma_crf-C_M * (T(t) - M_s))) + (1+xi_S0)/2)
%case 2
g12(t) = (2*xi_S0 -1) / (xi_S0-1) * Delta + (xi_S0-1 - Delta)/(xi_S0-1) * ((1-xi_S0)/2 * cos(pi/(sigma_crs-sigma_crf)*(sigma(t)-sigma_crf)+(1+xi_S0)/2))
%case 3
g13(t) = (xi_S0)/2 * (cos( a_A * ( T(t) - A_s - sigma(t)/C_A))+1)
the results is like this
>> g11
g11(t) =
((xi_S0 + xi_T0 - 1)*(xi_S0/2 - cos((pi*(sigma(t) - sigma_crf + C_M*(M_s - T(t))))/(sigma_crf - sigma_crs))*(xi_S0/2 - 1/2) + 1/2))/(xi_S0 - 1) - xi_T0/(xi_S0 - 1)
>> g12
g12(t) =
(Delta*(2*xi_S0 - 1))/(xi_S0 - 1) + (cos(xi_S0/2 + (pi*(sigma_crf - sigma(t)))/(sigma_crf - sigma_crs) + 1/2)*(xi_S0/2 - 1/2)*(Delta - xi_S0 + 1))/(xi_S0 - 1)
>> g13
g13(t) =
(xi_S0*(cos(a_A*(A_s - T(t) + sigma(t)/C_A)) + 1))/2 %the signal is changed
I'd like to ask if anyone know how to resolve this problem.
Best Rgds,
Kj Hu

Walter Roberson on 26 Oct 2020
Irrelevant. cos(-x) = cos(x) so cos(a*f(x)) = cos(a*(-f(x)))
K Hu on 26 Oct 2020
Edited: K Hu on 26 Oct 2020
The explication is very clear and thank you again to answer my question so quickly.
Since I don't have a large number of equations, I think I will do verification by myself.

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