Solution for two unknowns with two equations

[zoom1]

lb*cos(qb) - lc*cos(qc) = ld*cos(qd) - la*cos(qa)

lb*sin(qb) - lc*sin(qc) = ld*sin(qd) - la*sin(qa)

Those are the equations I have. I know all the parameters but qb and qc.

How can I solve for qb and qc ?

 

[zoki85]

lb*cos(qb) - lc*cos(qc) = ld*cos(qd) - la*cos(qa)

lb*sin(qb) - lc*sin(qc) = ld*sin(qd) - la*sin(qa)

Those are the equations I have. I know all the parameters but qb and qc.

How can I solve for qb and qc ?

This would mean the system is:

B⋅cos x - C⋅cos y = D
B⋅sin x - C⋅sin y = E

?

 

[mfb]

There is always the approach $\sin(x) = \sqrt{1-\cos^2(x)}$, but there might be a better solution for this particular set of equations.

 

[zoom1]

This would mean the system is:

B⋅cos x - C⋅cos y = D
B⋅sin x - C⋅sin y = E

?

Yes

 

[zoki85]

Transform it into system of algebraic equation with tangent half-angle substitution:
tan(x/2)=u ⇒ sin(x)=2u/(1+u²) , cos(x)=(1-u²)/(1+u²)
tan(y/2)=v ⇒ sin(y)=2v/(1+v²) , cos(y)=(1-v²)/(1+v²).

Unfortunatelly, it seems in general case the system can't be solved exactly,and you'll have to do it with aproximative numerical methods (iterativelly).

*Post edited for removing typos

 

[mfb]

It is possible to find an analytic solution. The common prefactors on the left side make it much easier.