We recall that $\mathscr{C}$ was defined as the image of the application $$\gamma : \left[\frac{\pi}{4}, \frac{7\pi}{4}\right] \rightarrow \mathbb{R}^2, t \mapsto (\cos t, 2\sin t)$$ In this question, we seek a complex parametrization of $\mathscr{C}$, of the form $$z : \left[\frac{\pi}{4}, \frac{7\pi}{4}\right] \rightarrow \mathbb{C}, t \mapsto \rho(t) \mathrm{e}^{\mathrm{i}\theta(t)}$$ where $\rho$ and $\theta$ are two continuous functions from $\left[\frac{\pi}{4}, \frac{7\pi}{4}\right]$ to $\mathbb{R}$, the function $\rho$ taking strictly positive values. III.B.1) Calculate $\rho(t)$ for all $t \in \left[\frac{\pi}{4}, \frac{7\pi}{4}\right]$. III.B.2) Represent on the calculator the parametrized arc $$\mathscr{G} : \left[\frac{\pi}{4}, \frac{7\pi}{4}\right] \rightarrow \mathbb{C}, t \mapsto \rho(t) \mathrm{e}^{\mathrm{i}t}$$ and reproduce the curve roughly on the paper. What letter does this curve evoke? III.B.3) From the expression of $\gamma(t)$, calculate $\tan\theta(t)$. III.B.4) a) Represent the function $t \mapsto \arctan(2\tan t)$ on the part of the interval $\left[\frac{\pi}{4}, \frac{7\pi}{4}\right]$ on which this function is defined. b) Modify this function to determine the continuous function $\theta$ sought. The result will be verified by representing with the aid of the calculator the parametrized curve $z$. III.B.5) Indicate a sequence of Maple or Mathematica instructions allowing one to obtain this plot.
We recall that $\mathscr{C}$ was defined as the image of the application
$$\gamma : \left[\frac{\pi}{4}, \frac{7\pi}{4}\right] \rightarrow \mathbb{R}^2, t \mapsto (\cos t, 2\sin t)$$
In this question, we seek a complex parametrization of $\mathscr{C}$, of the form
$$z : \left[\frac{\pi}{4}, \frac{7\pi}{4}\right] \rightarrow \mathbb{C}, t \mapsto \rho(t) \mathrm{e}^{\mathrm{i}\theta(t)}$$
where $\rho$ and $\theta$ are two continuous functions from $\left[\frac{\pi}{4}, \frac{7\pi}{4}\right]$ to $\mathbb{R}$, the function $\rho$ taking strictly positive values.
III.B.1) Calculate $\rho(t)$ for all $t \in \left[\frac{\pi}{4}, \frac{7\pi}{4}\right]$.\\
III.B.2) Represent on the calculator the parametrized arc
$$\mathscr{G} : \left[\frac{\pi}{4}, \frac{7\pi}{4}\right] \rightarrow \mathbb{C}, t \mapsto \rho(t) \mathrm{e}^{\mathrm{i}t}$$
and reproduce the curve roughly on the paper. What letter does this curve evoke?\\
III.B.3) From the expression of $\gamma(t)$, calculate $\tan\theta(t)$.\\
III.B.4)\\
a) Represent the function $t \mapsto \arctan(2\tan t)$ on the part of the interval $\left[\frac{\pi}{4}, \frac{7\pi}{4}\right]$ on which this function is defined.\\
b) Modify this function to determine the continuous function $\theta$ sought.\\
The result will be verified by representing with the aid of the calculator the parametrized curve $z$.\\
III.B.5) Indicate a sequence of Maple or Mathematica instructions allowing one to obtain this plot.