B2. Let $O$ be the origin of the coordinate system. We rotate the point $A\left(\frac{5}{2},-\frac{5 \sqrt{3}}{2}\right)$ around $O$ by $2013 \pi \mathrm{radians}$ to point $B$. We then reflect point $B$ over the bisector of the odd quadrants to point $C$. Calculate the measure of the angle $\Varangle A O C$.
B2. The coordinates of point $B$ are $\left(-\frac{5}{2}, \frac{5 \sqrt{3}}{2}\right)$, and the coordinates of point $C$ are $\left(\frac{5 \sqrt{3}}{2},-\frac{5}{2}\right)$. All three points lie on a circle with center $O$ and radius 5. From the values of the trigonometric functions $\sin \left(-30^{\circ}\right)=-\frac{1}{2}$ and $\cos \left(-30^{\circ}\right)=\frac{\sqrt{3}}{2}$, we understand that $O C$ subtends an angle of $30^{\circ}$ with the positive x-axis. Similarly, from $\sin \left(-60^{\circ}\right)=-\frac{1}{2}$ and $\cos \left(-60^{\circ}\right)=\frac{\sqrt{3}}{2}$, it follows that $O A$ subtends an angle of $60^{\circ}$ with the positive x-axis. From this, we calculate $\Varangle A O C=30^{\circ}$.

![](https://cdn.mathpix.com/cropped/2024_06_07_6340f9febbabfa1859cdg-21.jpg?height=557&width=557&top_left_y=1132&top_left_x=1298)

Writing the coordinates of point $B$

1 point

Writing the coordinates of point $C$

1 point

Determine that points $A, B$, and $C$ lie on a circle with center $O$ and radius 5

1 point

Justified determination that the line segment $O C$ subtends an angle of $30^{\circ}$ with the positive x-axis

1 point

Justified determination that the line segment $O A$ subtends an angle of $60^{\circ}$ with the positive x-axis

2nd method. Using the distance formula between two points, we calculate $|O A|=5$, $|O C|=5$, and $|A C|=\sqrt{25(2-\sqrt{3})}$. From this, using the cosine rule, we get

$$
\cos (\Varangle A O C)=\frac{|O A|^{2}+|O C|^{2}-|A C|^{2}}{2|O A||O C|}=\frac{25+25-25(2-\sqrt{3})}{100}=\frac{\sqrt{3}}{2},
$$

thus $\Varangle A O C=30^{\circ}$.
Calculated distances $|O A|$ and $|O C|$ ..... 1 point
Calculated distance $|A C|$ ..... 1 point
Written cosine rule ..... 1 point
Calculated $\cos (\Varangle A O C)$ ..... 2 points
Calculated angle $\Varangle A O C$ ..... 1 point