We define the applications: $$\alpha : \mathbb{N}^* \times \left[\frac{\pi}{4}, \frac{7\pi}{4}\right] \rightarrow \mathbb{R}, (n, t) \mapsto \frac{\pi}{4} + \frac{3\pi}{2n} \mathrm{E}\left(\frac{2n}{3\pi}\left(t - \frac{\pi}{4}\right)\right)$$ $$\omega : \mathbb{N}^* \times \left[\frac{\pi}{4}, \frac{7\pi}{4}\right] \rightarrow \mathbb{R}, (n, t) \mapsto \cos^2\left(\frac{2n}{3}\left(t - \frac{\pi}{4}\right)\right)$$ where $\mathrm{E}(x)$ denotes the integer part of the real number $x$. III.C.1) Briefly study $\alpha$ and $\omega$, then represent on the same graph the two functions $t \mapsto \alpha(10, t)$ and $t \mapsto \omega(10, t)$. III.C.2) Represent the function $\psi : \left[\frac{\pi}{4}, \frac{7\pi}{4}\right] \rightarrow \mathbb{R}, t \mapsto \frac{1}{4}\sin\left(\frac{2}{3}\left(t - \frac{\pi}{4}\right)\right)$. III.C.3) We define the function: $$w : \mathbb{N}^* \times \left[\frac{\pi}{4}, \frac{7\pi}{4}\right] \rightarrow \mathbb{C}, (n, t) \mapsto \rho(t)\left(1 + \psi(t)\omega(n, t)\right) \mathrm{e}^{\mathrm{i}\theta(\alpha(n, t))}$$ Identify which of the four graphics represents the function $t \mapsto w(40, t)$, and explain why. III.C.4) Write a sequence of Maple or Mathematica instructions allowing one to create the sequence of the first 100 curves (one may create an animation).
We define the applications:
$$\alpha : \mathbb{N}^* \times \left[\frac{\pi}{4}, \frac{7\pi}{4}\right] \rightarrow \mathbb{R}, (n, t) \mapsto \frac{\pi}{4} + \frac{3\pi}{2n} \mathrm{E}\left(\frac{2n}{3\pi}\left(t - \frac{\pi}{4}\right)\right)$$
$$\omega : \mathbb{N}^* \times \left[\frac{\pi}{4}, \frac{7\pi}{4}\right] \rightarrow \mathbb{R}, (n, t) \mapsto \cos^2\left(\frac{2n}{3}\left(t - \frac{\pi}{4}\right)\right)$$
where $\mathrm{E}(x)$ denotes the integer part of the real number $x$.
III.C.1) Briefly study $\alpha$ and $\omega$, then represent on the same graph the two functions $t \mapsto \alpha(10, t)$ and $t \mapsto \omega(10, t)$.\\
III.C.2) Represent the function $\psi : \left[\frac{\pi}{4}, \frac{7\pi}{4}\right] \rightarrow \mathbb{R}, t \mapsto \frac{1}{4}\sin\left(\frac{2}{3}\left(t - \frac{\pi}{4}\right)\right)$.\\
III.C.3) We define the function:
$$w : \mathbb{N}^* \times \left[\frac{\pi}{4}, \frac{7\pi}{4}\right] \rightarrow \mathbb{C}, (n, t) \mapsto \rho(t)\left(1 + \psi(t)\omega(n, t)\right) \mathrm{e}^{\mathrm{i}\theta(\alpha(n, t))}$$
Identify which of the four graphics represents the function $t \mapsto w(40, t)$, and explain why.\\
III.C.4) Write a sequence of Maple or Mathematica instructions allowing one to create the sequence of the first 100 curves (one may create an animation).