# Unknown angles as functions of time

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Dario Saracchi on 1 Jun 2021
Commented: William Rose on 3 Jun 2021
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?

William Rose on 1 Jun 2021
@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 .
##### 2 CommentsShowHide 1 older comment
William Rose on 3 Jun 2021
@Dario Saracchi, You're welcome. Yes.

R2021a

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