Math 6720 • Applied Complex Variables and Asymptotic Methods

Complete the following:
\begin{enumerate}
\item
(10 pts) Use de Moivre's Theorem to show that the two expressions,
\[
\cos(n\theta),\qquad \frac{\sin((n+1)\theta)}{\sin\theta},
\]
for arbitrary \(n\in\mathbb{N}\), are both degree-\(n\) polynomials of \(\cos\theta\).

\item
(10 pts) Prove the triangle inequality: Given \(z_1,z_2\in\mathbb{C}\), then
\[
\bigl||z_1|-|z_2|\bigr|\le |z_1+z_2|\le |z_1|+|z_2|.
\]
\end{enumerate}