Relationships in formula of rotation of a plane about a point $(a, b)$ and angle $\theta$

Question

Let $(a, b)$ be a point in $\mathbb{R}^2$ and $\theta$ be an angle of rotation. Using translation and roation matricies, namely $$\begin{pmatrix} 1 & 0 & a \\ 0 & 1 & b \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} \operatorname{cos}\theta & -\operatorname{sin}\theta & 0 \\ \operatorname{sin}\theta & \operatorname{cos}\theta & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 & -a \\ 0 & 1 & -b \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} x\\ y\\ 1 \end{pmatrix}$$ I found a formula for rotation a $(x, y)$ point around a set point $(a, b)$ about an angle $\theta$ that turned out to be $f(x, y) = (x\operatorname{cos}\theta - y\operatorname{sin}\theta -a\operatorname{cos}\theta + b\operatorname{sin}\theta + a, x\operatorname{sin}\theta + y\operatorname{cos}\theta -a\operatorname{sin}\theta + b\operatorname{cos}\theta + b)$. What made me curious about it is the fact that coefficients of $x, y, a, b$ in the second coordinate seem to be derivatives of coefficients of $x, y, a, b$ in the first coordinate. For example next to $x$ we have $\operatorname{cos}\theta$, but in the second coordinate we have $\operatorname{sin}\theta$ next to $x$, the same pattern goes for $y, a, b$. My question is: why is that? Is there some deeper connection between coordinates of rotation and translation? Or is it a mere coincidence? Reference on my knowledge is that I just finished undergraduate degree in mathematics.