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\begin{cases} 7cos(\theta_3) - 6cos(\theta_4) = 10.5 - 2.5cos(\theta_2) \\ 7jsin(\theta_3) - 6jsin(\theta_4) = - 2.5j sin(\theta_2) \end{cases}

This is a non-linear system of equations. There are an infinite number of solutions for theta_3 and theta_4. Even when we limit our range to [0,2π], there are 2 unique solutions, because the mechanism has a toggle position. To keep the solution consistent with the real world, all non-linear equations are solved numerically with initial conditions based on the measurements for the chair. In this case, the initial conditions and solutions are

$$ \theta_{3,guess} = 45 \hspace{1ex} degrees ; \theta_{4,guess} = 120 \hspace{1ex} degrees ; \theta_{2,t=0} = 165 \hspace{1ex} degrees $$
\begin{cases} \theta_{3,t=0} = 2.6 \hspace{1ex} degrees \\ \theta_{4,t=0} = 170.8 \hspace{1ex} degrees \end{cases}