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

$\theta_{\phi} = e^{\frac{i\times\theta}{2} \times (\cos(\phi)\times \sigma_x + \sin(\phi) \times \sigma_y)}$

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?