I'm having trouble understanding the polar representation of quaternions. That is, any quaternion $z=a+ib+jc+kd=a+\mathbf{v}$ can be expressed in polar form as: $$ z = |z|\left(\cos \theta +\mathbf{n}\sin \theta \right) $$ with: $$ \begin{cases} & |z|= \sqrt{a^2+b^2+c^2+d^2}\\ & \cos \theta=\dfrac{a}{|z|} \qquad \sin \theta=\dfrac{|\mathbf{v}|}{|z|}\\ & \mathbf{n}=\dfrac{\mathbf{v}}{|z|\sin \theta}=\dfrac{\mathbf{v}}{|\mathbf v|} \end{cases} $$

In particular, I'm having trouble with the formula for the exponentiation of a quaternion in this form: $$ z^n = |z|^n\left(\cos n\theta +\mathbf{n}\sin n\theta \right) $$ I have tried exponentiating an arbitrary quaternion $a+ib+jc+kd$ by a power of 2 in hopes of understanding what exactly is going on here, but I can't seem to obtain the $\sin 2\theta$ and $\cos 2\theta$ terms. I understand why such a $\theta$ exists such that you can assign $$ \cos \theta=\dfrac{a}{|z|} \qquad \sin \theta=\dfrac{|\mathbf{v}|}{|z|} $$ However, defined in this way, I don't know what the expressions $\sin n \theta$ and $\cos n \theta$ would represent, especially in the context of quaternions. If anyone could shed some light on this I would be extremely appreciative!