2.2. The coordinate system $X O Y$ was first translated in parallel so that the new origin became located at point $O^{\prime}(2,3)$, and then the axes were rotated by an angle $\varphi=30^{\circ}$. Determine the new coordinates of point $A$, if the old coordinates in the system $X O Y$ are: $A(-1,5)$. Determine the coordinates that point $B$ had in the old system XOY, if in the new coordinate system it is given as: $B(4,-2)$.
Solution:

a) when the coordinate system is translated, the coordinates of point $A$ will be as follows:

$$
\left\{\begin{array}{l}
x_{A}^{\prime}=x_{A}-a=-1-2=-3 \\
y_{A}^{\prime}=y_{A}-b=5-3=2
\end{array}\right.
$$

when the axes are rotated by an angle of $30^{\circ}$, the coordinates of point $A$ are transformed as follows:

$$
\begin{gathered}
x_{A}^{\prime \prime}=x_{A}^{\prime} \cos \varphi+y_{A}^{\prime} \sin \varphi=(-3) \cos 30^{\circ}+2 \sin 30^{\circ}= \\
=-3 \cdot \frac{1}{2}+2 \cdot \frac{\sqrt{3}}{2}=\frac{2 \sqrt{3}-3}{2} \approx 0.722 \\
y_{A}^{\prime \prime}=-x_{A}^{\prime} \sin \varphi+y_{A}^{\prime} \cos \varphi=-(-3) \sin 30^{\circ}+2 \cos 30^{\circ}= \\
=3 \cdot \frac{\sqrt{3}}{2}+2 \cdot \frac{1}{2} \approx 3.598
\end{gathered}
$$

b) in the coordinate system $X O Y$, point $B$ has the following coordinates:

$$
\begin{aligned}
& x_{B}^{\prime}=x_{B}^{\prime \prime} \cos \varphi-y_{B}^{\prime \prime} \sin \varphi=4 \cos 30^{\circ}-(-2) \sin 30^{\circ} \approx 3.732 \\
& y_{B}^{\prime}=x_{B}^{\prime \prime} \sin \varphi+y_{B}^{\prime} \cos \varphi=4 \sin 30^{\circ}+(-2) \cos 30^{\circ} \approx 2.464
\end{aligned}
$$

In the coordinate system $X O Y$, point $B$ has the coordinates:

$$
x_{B}=x_{B}^{\prime}+a=3.732+2 \approx 5.732 ; \quad y_{B}=y_{B}^{\prime}+b \approx 5.464
$$

Answer: $x_{A}^{\prime \prime} \approx 0.722 ; y_{A}^{\prime \prime} \approx 3.598 ; x_{B} \approx 5.732 ; y_{B} \approx 5.464$.