How to tell if a set of matrices generates $SL_2(\mathbb{R})$?

Question

It is fairly straightforward to show that if $N$ is the set of upper triangular matrices with ones on the diagonal and $\gamma$ is the rotation matrix by $\pi/2$ radians then $SL_2(\mathbb{R})=\langle \gamma, N\rangle$. However, what if we have for example that $\gamma$ is the rotation by $\pi/3$ radians? Or some other non-trivial rotation matrix (i.e. not $\pm I$)? What if $\gamma$ is something simply conjugate to a non-trivial rotation? My intuition says all these cases should at the very least generate a dense set in $SL_2(\mathbb{R})$ but I see no way of getting there.