The complex plane is given an orthonormal direct coordinate system ($\mathrm{O}; \vec{u}, \vec{v}$). The unit of length is one centimetre.

\begin{enumerate}
  \item Solve in $\mathbb{C}$ the equation $\left(z^{2} - 2z + 4\right)\left(z^{2} + 4\right) = 0$.
  \item We consider the points A and B with complex numbers $z_{\mathrm{A}} = 1 + \mathrm{i}\sqrt{3}$ and $z_{\mathrm{B}} = 2\mathrm{i}$ respectively.\\
a. Write $z_{\mathrm{A}}$ and $z_{\mathrm{B}}$ in exponential form and justify that the points A and B lie on a circle with centre O, whose radius you will specify.\\
b. Draw a figure and place the points A and B.\\
c. Determine a measure of the angle $(\overrightarrow{\mathrm{OA}}, \overrightarrow{\mathrm{OB}})$.
  \item We denote by F the point with complex number $z_{\mathrm{F}} = z_{\mathrm{A}} + z_{\mathrm{B}}$.\\
a. Place the point F on the previous figure. Show that OAFB is a rhombus.\\
b. Deduce a measure of the angle $(\overrightarrow{\mathrm{OA}}, \overrightarrow{\mathrm{OF}})$ then of the angle $(\vec{u}, \overrightarrow{\mathrm{OF}})$.\\
c. Calculate the modulus of $z_{\mathrm{F}}$ and deduce the expression of $z_{\mathrm{F}}$ in trigonometric form.\\
d. Deduce the exact value of:
$$\cos\left(\frac{5\pi}{12}\right)$$
  \item Two calculator models from different manufacturers give for one:
$$\cos\left(\frac{5\pi}{12}\right) = \frac{\sqrt{2 - \sqrt{3}}}{2}$$
and for the other:
$$\cos\left(\frac{5\pi}{12}\right) = \frac{\sqrt{6} - \sqrt{2}}{4}$$
Are these results contradictory? Justify your answer.
\end{enumerate}