CIS question: proving rotation/multiplication relation

[Trenters4325]

I understand that when you multiply a coordinate in the complex plane by cis(theta), that point is rotated theta degrees clockwise. I'm trying to prove this though, and I'm having trouble. Can someone help?

 

[Matt]

Let \(\displaystyle \L z=r\text{cis}(\alpha)\) be any point on the complex plane.

Then:

\(\displaystyle \L \quad\begin{eqnarray}
z\cdot\text{cis}\theta
&=&r\text{cis}\alpha\text{cis}\theta\\
&=&r(\cos\alpha+i\sin\alpha)(\cos\theta+i\sin\theta)\\
&=&r[(\cos\alpha\cos\theta-\sin\alpha\sin\beta)+i(\cos\alpha\sin\theta+\cos\theta\sin\alpha)]
\end{eqnarray}\)

Can you finish it?

 

[Trenters4325]

Those are the right sides of the angle sum identities for cosine and sine, right?