We want to model in the plane the shell of a nautilus using a broken line in the form of a spiral. We are interested in the area delimited by this line.

We equip the plane with a direct orthonormal coordinate system $(O; \vec{u}; \vec{v})$. Let $n$ be an integer greater than or equal to 2. For all integer $k$ ranging from 0 to $n$, we define the complex numbers $z_k = \left(1 + \dfrac{k}{n}\right) \mathrm{e}^{\mathrm{i}\frac{2k\pi}{n}}$ and we denote by $M_k$ the point with affix $z_k$. In this model, the perimeter of the nautilus is the broken line connecting all the points $M_k$ with $0 \leqslant k \leqslant n$.

\textbf{Part A: Broken line formed from seven points}

In this part, we assume that $n = 6$. Thus, for $0 \leqslant k \leqslant 6$, we have $z_k = \left(1 + \dfrac{k}{6}\right) \mathrm{e}^{\mathrm{i}\frac{2k\pi}{6}}$.

\begin{enumerate}
  \item Determine the algebraic form of $z_1$.
  \item Verify that $z_0$ and $z_6$ are integers that you will determine.
  \item Calculate the length of the altitude from $M_1$ in the triangle $OM_0M_1$ then establish that the area of this triangle is equal to $\dfrac{7\sqrt{3}}{24}$.
\end{enumerate}

\textbf{Part B: Broken line formed from $n+1$ points}

In this part, $n$ is an integer greater than or equal to 2.

\begin{enumerate}
  \item For all integer $k$ such that $0 \leqslant k \leqslant n$, determine the length $OM_k$.
  \item For $k$ an integer such that $0 \leqslant k \leqslant n-1$, determine a measure of the angles $(\vec{u}; \overrightarrow{OM_k})$ and $(\vec{u}; \overrightarrow{OM_{k+1}})$. Deduce a measure of the angle $(\overrightarrow{OM_k}; \overrightarrow{OM_{k+1}})$.
  \item For $k$ an integer such that $0 \leqslant k \leqslant n-1$, calculate the area of the triangle $OM_kM_{k+1}$ as a function of $n$ and $k$.
\end{enumerate}