$$R_x(\theta) = e^{-i\theta X/2}$$ $$R_y(\theta) = e^{-i\theta Y/2}$$ $$R_z(\theta) = e^{-i\theta Z/2}$$

After applying the exponential map $e^A = 1 + A + A^2 / 2 + \dots$ and $X^2 = Y^2 = Z^2 = I$:

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