# Rotate $$$\left(3 \sqrt{2}, \frac{\sqrt{2}}{2}\right)$$$ by $$$45^{\circ}$$$ counterclockwise around $$$\left(0, 0\right)$$$

### Your Input

**Rotate $$$\left(3 \sqrt{2}, \frac{\sqrt{2}}{2}\right)$$$ by the angle $$$45^{\circ}$$$ counterclockwise around $$$\left(0, 0\right)$$$.**

### Solution

Rotation of a point $$$\left(x, y\right)$$$ around the origin by the angle $$$\theta$$$ counterclockwise will give a new point $$$\left(x \cos{\left(\theta \right)} - y \sin{\left(\theta \right)}, x \sin{\left(\theta \right)} + y \cos{\left(\theta \right)}\right)$$$.

In our case, $$$x = 3 \sqrt{2}$$$, $$$y = \frac{\sqrt{2}}{2}$$$, and $$$\theta = 45^{\circ}$$$.

Therefore, the new point is $$$\left(3 \sqrt{2} \cos{\left(45^{\circ} \right)} - \frac{\sqrt{2}}{2} \sin{\left(45^{\circ} \right)}, 3 \sqrt{2} \sin{\left(45^{\circ} \right)} + \frac{\sqrt{2}}{2} \cos{\left(45^{\circ} \right)}\right) = \left(\frac{5}{2}, \frac{7}{2}\right).$$$

### Answer

**The new point is $$$\left(\frac{5}{2}, \frac{7}{2}\right) = \left(2.5, 3.5\right)$$$A.**