Consider the points where $x\in X_1=\{(4k+1)\pi/2:\ k\in\mathbb{Z}\}$. For any $x\in X_1$, $\sin x=1$. Now consider any $t\in \mathbb R$; there exists $k\in\mathbb Z$ such that $(4k+1)\pi/2\leq t<(4(k+1)+1)\pi/2$. Then there exist $\alpha\in[0,1]$ with $t=\alpha(4k+1)\pi/2+(1-\alpha)(4(k+1)+1)\pi/2$. Thus $ \begin{bmatrix}t\\1\end{bmatrix}=\alpha\begin{bmatrix}(4k+1)\pi/2\\1\end{bmatrix}+(1-\alpha)\begin{bmatrix}(4(k+1)+1)\pi/2\\1\end{bmatrix}. $ This shows that $\begin{bmatrix}t\\1\end{bmatrix}$ is in the convex hull of $S$ for all $t\in \mathbb R$ (i.e. the whole line $y=1$ is).
Similarly, working with the numbers $(4k+3)\pi/2$ we get that $\begin{bmatrix}t\\-1\end{bmatrix}$ is in the convex hull for all $t$.
Now take $x\in\mathbb R$, $y\in[-1,1]$. Then there exists $\alpha\in[0,1]$ such that $y=\alpha\,1+(1-\alpha)(-1)$. Then $ \begin{bmatrix}x\\ y\end{bmatrix}=\alpha\begin{bmatrix}x\\1\end{bmatrix}+(1-\alpha)\begin{bmatrix}x\\-1\end{bmatrix}. $
In other words, $ \text{conv}\,S=\left\{\begin{bmatrix}x\\ y\end{bmatrix}:\ x\in\mathbb R,\ y\in[-1,1]\right\}. $