# How to solve simultaneous equation with trigonometric function

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christina widyaningtyas on 26 Jul 2022
Commented: Torsten on 30 Jul 2022
Hello, any one can help me
syms alpha gamma R d
R = 0.08;
d = 1.4;
R^2 == 4*(alpha)^2 / [(2*(alpha)^2 + 1) * acos*(2*gamma) + (2*(alpha)^2 - 1) * cos*(2*gamma) + 2*alpha*(asin*(2*gamma)) - sin*(2*gamma)]
d == atan*[(atan(2*alpha*tan*gamma + 1) + tan*gamma) / (tan*gamma - atan*gamma + 2*alpha)]
sol = solve ([R^2, d] , [alpha, gamma])
alphaSol = sol.alpha
gammaSol = sol.gamma
I need to obtain alpha and gamma
There is : Error using acos
Not enough input arguments.
• if I delete acos, another error is appear (cos, ect)

KSSV on 26 Jul 2022
You cannot write acos*(2*gamma) and tan*gamma. Note that they are function and they need someinput. It migh be: acos(2*gamma) and tan(gamma)
syms alpha gamma R d
R = 0.08;
d = 1.4;
R^2 == 4*(alpha)^2 / ((2*(alpha)^2 + 1) * acos(2*gamma) + (2*(alpha)^2 - 1) * cos(2*gamma) + 2*alpha*(asin(2*gamma)) - sin(2*gamma))
d == atan((atan(2*alpha*tan(gamma) + 1) + tan(gamma)) / (tan(gamma) - atan(gamma) + 2*alpha))
sol = solve ([R^2, d] , [alpha, gamma])
alphaSol = sol.alpha
gammaSol = sol.gamma
Though the above is not giving any solution, you need to check your expression properly.
Torsten on 30 Jul 2022
Seems there are several solutions to your equations:
R = 0.07;
d = -135;
fun = @(alpha,gamma)[4*alpha^2/((2*alpha^2+1)*cosh(2*gamma)+(2*alpha^2-1)*cos(2*gamma)+2*alpha*(sinh(2*gamma)-sin(2*gamma)))-R^2;...
(tanh(2*alpha*tan(gamma)+1)+tan(gamma))/(tan(gamma)-tanh(gamma)+2*alpha)-tand(d)];
options = optimset('TolFun',1e-8,'TolX',1e-8);
alpha0 = 0.15;
gamma0 = 1.7;
x0 = [alpha0,gamma0];
sol = fsolve(@(x)fun(x(1),x(2)),x0,options);
Equation solved. fsolve completed because the vector of function values is near zero as measured by the value of the function tolerance, and the problem appears regular as measured by the gradient.
alpha = sol(1)
alpha = 0.1786
gamma = sol(2)
gamma = 1.7811
fun(alpha,gamma)
ans = 2×1
1.0e-11 * 0.0055 -0.7673
alpha0 = -0.4;
gamma0 = -2.3;
x0 = [alpha0,gamma0];
sol = fsolve(@(x)fun(x(1),x(2)),x0,options);
Equation solved. fsolve completed because the vector of function values is near zero as measured by the value of the function tolerance, and the problem appears regular as measured by the gradient.
alpha = sol(1)
alpha = -0.3679
gamma = sol(2)
gamma = -2.3475
fun(alpha,gamma)
ans = 2×1
1.0e-08 * -0.0013 0.1872
syms alpha gamma %R d
R = 0.07;
d = -135;
expr1 = R^2 == 4*alpha^2/((2*alpha^2+1)*cosh(2*gamma)+(2*alpha^2-1)*cos(2*gamma)+2*alpha*(sinh(2*gamma)-sin(2*gamma)));
expr2 = tand(d) == (tanh(2*alpha*tan(gamma)+1)+tan(gamma))/(tan(gamma)-tanh(gamma)+2*alpha);
[sol_alpha, sol_gamma] = solve([expr1,expr2], [alpha,gamma])
Warning: Unable to solve symbolically. Returning a numeric solution using vpasolve.
sol_alpha =
sol_gamma =
fun(double(sol_alpha),double(sol_gamma))
ans = 2×1
1.0e-15 * -0.0017 0.4441