Show that exponential of a Pauli matrix $A$ can be simplified as follows:
$$ \exp(i A \theta) = \cos(\theta) I + i \sin(\theta) A $$
Let's separate even and odd powers of $\theta$ in our sum. $$ \exp(iA\theta) = (1 - \frac{\theta^2}{2!} + \frac{\theta^4}{4!} + \cdots )I $$ $$ + (\frac{\theta}{1!} - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} + \cdots)iA $$
Do these even and odd terms remind you of any special functions?