Mathematica says
$\Large \theta = \tan ^{-1}\left(\frac{-\sqrt{L_1^2 R_2^2-L_2^2 R_2^2+R_2^4}-L_1 L_2}{L_1^2+R_2^2},\frac{-\frac{L_2 L_1^2}{L_1^2+R_2^2}-\frac{L_1 \sqrt{-R_2^2 \left(-L_1^2+L_2^2-R_2^2\right)}}{L_1^2+R_2^2}+L_2}{R_2}\right)$
where the $\tan^{-1}(x,y)$ function takes two arguments and is quadrant aware.
To solve an equation of the form
$\displaystyle r \sin \theta =a+b \cos \theta $
square both sides and let $\displaystyle c=\cos \theta$ to get a quadratic equation in $\displaystyle c$
$\displaystyle \left(b^2+r^2\right)c^2+2 a b c +\left(a^2-r^2\right) =0$
where $\displaystyle \frac{\sqrt{2}}{2}<c<1$
The discriminant must be positive so a necessary condition is that $\displaystyle r^2\geq a^2-b^2$