$$R_x(\theta) = e^{-i\theta\sigma_x/2}$$

After applying the exponential map:

$$R_x(\theta) = \cos\left(\frac{\theta}{2}\right)I + i\sin\left(\frac{\theta}{2}\right)X$$ $$R_y(\theta) = \cos\left(\frac{\theta}{2}\right)I + i\sin\left(\frac{\theta}{2}\right)Y$$ $$R_z(\theta) = \cos\left(\frac{\theta}{2}\right)I + i\sin\left(\frac{\theta}{2}\right)Z$$