This works out nicely almost-automatically with groebner basis. Your single equation is equivalent to the two equations
$s^2+c^2=1$ $3s+4c=2$
The groebner basis in lexicographical order is
$4c+3s-2 ~~\textrm{and}~~ 25s^2-12s-12$
So solving the quadratic equation $25s^2-12s-12=0$ gives us the solutions! $s_{1,2}=\frac{6}{25}\frac{+}{-}\frac{4}{25}\sqrt{21}$ and $c_{1,2}=(2-3s_{1,2})/4$. Then $x_{1,2}=\arcsin(s_{1,2})=\arccos(c_{1,2})$. However, if you have some software to do groebner basis for you, then it probably already has a solve method to use directly on your first equation.