Unknown angles as functions of time

I have 2 functions in which the unknown variables are funcions of time, here are the equations:

AB=150;

AEh=90+sqrt(250^2-180^2);

k2=(150*sin(3*pi/4));

BE=sqrt((AEh+k2)^2+(k2)^2);

CE=250;

AC=sqrt(90^2+180^2);

delta=atan(2);

t_vec=0:0.1:3*pi;

alfa=3/4*pi+1/9*pi*sin(t_vec);

AB*cos(alfa)+BE*cos(beta)+CE*cos(gamma)-AC*cos(delta)=0

AB*sin(alfa)+BE*sin(beta)+CE*sin(gamma)-AC*sin(delta)=0

(beta and gamma are unknown variables)

I tried to use the syms function but it didn't work.

syms beta gamma
    eq=[AB*cos(alfa)+BE*cos(beta)+CE*cos(gamma)-AC*cos(delta)==0,AB*sin(alfa)+BE*sin(beta)+CE*sin(gamma)-AC*sin(delta)==0];
    sol=solve(eq);
    betasol=vpa(sol.beta)
    gammasol=vpa(sol.gamma)

How do I write a code to find beta and gamma?

Answer 1

[William Rose]

@Dario Saracchi, You can do some algebra before you do some code.

Define some constants:

d=AB*cos(alfa)-AC*cos(delta)

e=AB*sin(alfa)-AC*sin(delta)

a=BE, c=CE.

Then the equations

AB*cos(alfa)+BE*cos(beta)+CE*cos(gamma)-AC*cos(delta)=0

AB*sin(alfa)+BE*sin(beta)+CE*sin(gamma)-AC*sin(delta)=0

can be written

Now define and . Then the equations above can be written as

Therefore , and therefore

which is one equation and one unknown. Try solving this with solve() or with vpasolve() or numerically with fzero(). Then you can compute and .

[Dario Saracchi]

Thank you so much! And to plot Beta and Gamma as functions of time can I just use the plot function?

[William Rose]

@Dario Saracchi, You're welcome. Yes.