How to solve simultaneous equation with trigonometric function

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)

Answer 1

[KSSV]

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.

[christina widyaningtyas]

But, the results is empty @KSSV

alphaSol =

Empty sym: 0-by-1

gammaSol =

Empty sym: 0-by-1

[KSSV]

I already mentioned that. Check your expression. REad about solve. You can give some assumptions to your solution if you have.

[Torsten]

syms alpha gamma %R d 
R = 0.08;
d = 1.4;
expr1 = 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))                   

expr1 = 

expr2 = d == atan((atan(2*alpha*tan(gamma) + 1) + tan(gamma)) / (tan(gamma) - atan(gamma) + 2*alpha))

expr2 = 

sol = solve ([expr1,expr2] , [alpha, gamma])

Warning: Unable to solve symbolically. Returning a numeric solution using vpasolve.

sol = struct with fields:
    alpha: 0.89004376539962278983379602051241 + 0.38919789229051247720270451525197i
    gamma: 78.222544477417321649984876860727 - 3.0960897467326888301848990655128i

alphaSol = sol.alpha

alphaSol = 

gammaSol = sol.gamma 

gammaSol = 

[christina widyaningtyas]

Thank you so much @Torsten, its works

[christina widyaningtyas]

Dear @Torsten

I apologize in advance, it looks like I made a mistake by writing the equation at first.

The equation I mean as attached,

I don't know why I can't check it into that form.

If you don't mind, can you check my writing equation :

expr1 = (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)))

is it same with the attached equation or not, please?

If I use live editor with latex equation, it's same,

|R|^2 = \dfrac{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)}

but I haven't been successful to use it in the live editor, so I wrote it manually.

Thank you in advance

[Torsten]

Note that tanh, sinh and cosh are the hyperbolic tangent, hyperbolic sine and hyperbolic cosine, not atan, asin and acos which are the inverse tangent, inverse sin and inverse cosine functions.

syms alpha gamma %R d 
R = 0.08;
d = 1.4;
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)))

expr1 = 

expr2 = d == atan((tanh(2*alpha*tan(gamma)+1)+tan(gamma))/(tan(gamma)-tanh(gamma)+2*alpha))

expr2 = 

sol = solve ([expr1,expr2] , [alpha, gamma])

Warning: Unable to solve symbolically. Returning a numeric solution using vpasolve.

sol = struct with fields:
    alpha: 0.74245496926973860397234179524781
    gamma: 2.627014792787395537585234942901

alphaSol = sol.alpha

alphaSol = 

0.74245496926973860397234179524781

gammaSol = sol.gamma 

gammaSol = 

2.627014792787395537585234942901

[christina widyaningtyas]

Thank you so much @Torsten

I still trying for others R and d values,

R = 0.022 to 0.12

d = 60 to 150

but the results are :

alphaSol = Empty sym: 0-by-1

gammaSol = Empty sym: 0-by-1

If I read, they said it might be because there isn't any solution to the equation

[Torsten]

atan is between -pi/2 and pi/2. So I don't understand how "d" could be between 60 and150 - except you want to get complex solutions, maybe.

[christina widyaningtyas]

actually d is phase difference in degree

Last time I use in radians, but it should be in degree

Does that mean it can only be solved with complex solution @Torsten?

[Torsten]

In your test equation, d was 7/5 which does not seem to be in degrees.

However: if you want to prescribe d in degrees, change

expr2 = d == atan((tanh(2*alpha*tan(gamma)+1)+tan(gamma))/(tan(gamma)-tanh(gamma)+2*alpha))

to

expr2 = d == atand((tanh(2*alpha*tan(gamma)+1)+tan(gamma))/(tan(gamma)-tanh(gamma)+2*alpha))

[christina widyaningtyas]

I already tried that one @Torsten

I checked with substituting R = 0.07 and d = -135, they will give alpha = 0.15 and gamma 1.7

but, the results still

alphaSol =

Empty sym: 0-by-1

gammaSol =

Empty sym: 0-by-1

:((((((

[Torsten]

Your equations cannot be solved symbolically for alpha and gamma. That's why MATLAB uses vpasolve to return a numerical solution.

[christina widyaningtyas]

I tried :

%%

syms alpha gamma %R d

R = 0.07;

d = -135;

[sol_alpha, sol_gamma] = vpasolve([(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))), d == atand((tanh(2*alpha*tan(gamma)+1)+tan(gamma))/(tan(gamma)-tanh(gamma)+2*alpha))], [alpha,gamma])

But :

• Undefined function 'atand' for input arguments of type 'sym'.

then, when I change atand to atan back to eror : sol_alpha = Empty sym: 0-by-1 ; sol_gamma = Empty sym: 0-by-1

[Alex Sha]

Hi, Christina, if your equations are:

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))

The results will be:

alpha: 0.0503567460786745
gamma: -0.234530822330344

While if your eauqtions are:

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))                   
(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))) 

then there will not be solution in real data type

[christina widyaningtyas]

Thank you for your comment @Alex Sha

I run code :

%%

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 = d == atan((tanh(2*alpha*tan(gamma)+1)+tan(gamma))/(tan(gamma)-tanh(gamma)+2*alpha))

sol = solve ([expr1,expr2] , [alpha, gamma])

alphaSol = sol.alpha

gammaSol = sol.gamma

also :

%%

syms alpha gamma %R d

R = 0.07;

d = -135;

[sol_alpha, sol_gamma] = vpasolve([(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))), d == atand((tanh(2*alpha*tan(gamma)+1)+tan(gamma))/(tan(gamma)-tanh(gamma)+2*alpha))], [alpha,gamma])

I tried the code by substituting R (0.07) and d (-135 in degress) to check my code, it should be give alpha = 0.15 and gamma = 1.7

But still gives an errors

[Walter Roberson]

Take the tand out of both sides of the second equation, so

tand(d) == tand(atand(Expression))

and simplify to

tand(d) == Expression

it makes it easier for vpasolve to find solutions

[Torsten]

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