r2+r3=r1+r4(1)

r2*exp(i*theta2)+r3*exp(i*theta3)=r1*exp(i*theta1)+r4*exp(i*theta4) (2)

Note link lengths r1,r2,r3, and r4 along with theta1 are constants. Let theta2 be the independent variable, and theta3 and theta4 be the dependent variables. Rearanging the equation, we have

r3*exp(i*theta3)-r4*exp(i*theta4)=r1*exp(i*theta1)-r2*exp(i*theta2). (3)

Let R1=r3, phi1=theta3, R2=-r4, phi2=theta4, and z=(x3,y3)=r1*exp(i*theta1)-r2*exp(i*theta2). We now have a general complex equation

R1*exp(i*phi1)+R2*exp(i*phi2)=z. (4)

Angular positions theta3 and theta4 can now be solved for given parameters r1,r2,r3,r4,theta1, and theta2. From equation (4) we obtain

cos(phi1)=(x3-R2cos(phi2))/R1 (5)

sin(phi1)=(y3-R2sin(phi2))/R1. (6)

Substituting these results into the trig identity sin^2(phi1)+cos^2(phi1)=1 and simplifying we obtain

y3*sin(phi2)+x3*cos(phi2)=(x3^2+y3^2+R2^2-R1^2)/2*R2. (7)

From this equation we can obtain formulas for phi1 and phi2

phi2=atan2(y3,x3)± acos((x3^2+y3^2+R2^2-R1^2)/2*R2*sqrt(x3^2+y3^2)) (8)

phi1=atan2(sin(phi1),cos(phi1)) =atan2((y3-R2sin(phi2))/R1, (x3-R2cos(phi2))/R1). (9)

Similar equations can be derived assuming either theta3 or theta4 is known with the other two angles as parameters.

Please enter link lengths, theta1 and one other known angle to find the other two angles.