So I have the following problem. Coordinates of the red coordinate system are given. Also coordinates of point $T$ are known (expressed in the black CS).
What I would need is to write the coordinates of point $T$ in red CS assuming the rotation angle between the y and y' axes is $\varphi$.
Since the local coordinate system (red) is rotated CW by $\varphi$ the 2×2 rotation matrix (local to world) is
$$\mathrm{E} = \left| \matrix{ \cos\varphi & \sin\varphi \\ -\sin\varphi & \cos\varphi } \right| $$
In terms of the vectors $\mathbf{r}$ in the world coordinates and $\mathbf{r}'$ in the local coordinates you have
$$\mathbf{r} = \mathbf{R} + \mathrm{E} \, \mathbf{r}' $$
$$ \pmatrix{x\\y} = \pmatrix{R_x\\R_y} + \left| \matrix{ \cos\varphi & \sin\varphi \\ -\sin\varphi & \cos\varphi } \right| \pmatrix{x' \\ y'} $$
So you need to solve for $\mathbf{r}'$, or
$$ \pmatrix{x'\\y'} =\left| \matrix{ \cos\varphi & -\sin\varphi \\ \sin\varphi & \cos\varphi } \right| \left( \pmatrix{x \\ y} -\pmatrix{R_x\\R_y} \right)$$
by component, the above is
$$ \begin{aligned} x' &= \cos\varphi ( x-R_x) - \sin\varphi (y-R_y) \\ y' & =\sin\varphi( x-R_y) + \cos\varphi (y-R_y) \end{aligned}$$
Where $\mathbf{R} = \pmatrix{R_x \\ R_y}$ are the coordinates of the origin of the red (local) system in the black (world) coordinates.