
Faisons un schéma:
\begin{center}
\begin{tikzpicture}
\draw[->,>=latex,thick] (-1,0) -- (5,0);
\draw[->,>=latex,thick] (0,-1) -- (0,5);
\node[below left] at (0,0) {$O$};
\fill[red] (4,1) circle(1pt) node[above right] {$M$};
\fill[blue] ({4*cos(40)-sin(40)},{4*sin(40)+cos(40)}) circle(1pt) node[above left] {$M'$};
\draw[red] (0,0) -- (4,1);
\draw[blue] (0,0) -- ({4*cos(40)-sin(40)},{4*sin(40)+cos(40)});
\draw[->,>=latex,red] (3,0) arc(0:14:3);
\draw[->,>=latex,blue] (14:2) arc(14:54:2);
\node[right,red] at (7:3) {$\alpha$};
\node[right,blue] at (34:2) {$\theta$};
\draw[dashed,red] (4,0) node[below] {$x$} -- (4,1) -- (0,1) node[left] {$y$};
\draw[dashed,blue] ({4*cos(40)-sin(40)},0) node[below] {$x'$} -- ({4*cos(40)-sin(40)},{4*sin(40)+cos(40)}) -- (0,{4*sin(40)+cos(40)}) node[left] {$y'$};
\end{tikzpicture}
\end{center}

Dans le triangle rectangle rouge, on a:
\[
\begin{cases}
x = OM\cos\alpha\\
y = OM\sin\alpha
\end{cases}.\tag{1}
\]
Dans le triangle rectangle bleu, on a:
\[
\begin{cases}
x' = OM'\cos(\alpha+\theta) = OM\big(\cos\alpha\cos\theta-\sin\alpha\sin\theta \big)\\
y' = OM'\sin(\alpha+\theta) = OM\big( \sin\alpha\cos\theta+\sin\theta\cos\alpha \big)
\end{cases}
\]
soit:
\[
\begin{cases}
x' = (OM\cos\alpha)\cos\theta - (OM\sin\alpha)\sin\theta\\
y' = (OM\sin\alpha)\cos\theta + (OM\cos\alpha)\sin\theta
\end{cases}
\]
et donc, en tenant compte des égalités (1):
\[
\begin{cases}
x' = x\cos\theta - y\sin\theta\\
y' = x\sin\theta + y\cos\theta
\end{cases}
\]
soit:
\[
\dbinom{x'}{y'} = \begin{pmatrix}x\cos\theta-y\sin\theta\\x\sin\theta+y\cos\theta\end{pmatrix} = \begin{pmatrix}\cos\theta & -\sin\theta\\\sin\theta & \cos\theta\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}  = R\dbinom{x}{y}.
\]
