Mathematical representation of rotation around bloch sphere

I learned a formula to represent the rotation around bloch sphere:

${\theta }_{\varphi }=e^{\frac{i\times \theta }{2}\times (\cos (\varphi )\times {\sigma }_x+\sin (\varphi )\times {\sigma }_y)}$

So that ${\pi }_0$ is the gate $X$ and ${\pi }_{\frac{\pi }{2}}$ is the gate $Y$.

My question is: how do we get this formula to represent the rotation? If we have a general rotation, say ${\pi }_{\frac{\pi }{6}}$, how do we use the rotation gates like $R_x$, $R_y$, or $R_Z$ to represent the gate?