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Rotate $$$\left(2 \sqrt{2}, \frac{\sqrt{2}}{2}\right)$$$ by $$$45^{\circ}$$$ counterclockwise around $$$\left(0, 0\right)$$$
Your Input
Rotate $$$\left(2 \sqrt{2}, \frac{\sqrt{2}}{2}\right)$$$ 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, $$$x = 2 \sqrt{2}$$$, $$$y = \frac{\sqrt{2}}{2}$$$, and $$$\theta = 45^{\circ}$$$.
Therefore, the new point is $$$\left(2 \sqrt{2} \cos{\left(45^{\circ} \right)} - \frac{\sqrt{2}}{2} \sin{\left(45^{\circ} \right)}, 2 \sqrt{2} \sin{\left(45^{\circ} \right)} + \frac{\sqrt{2}}{2} \cos{\left(45^{\circ} \right)}\right) = \left(\frac{3}{2}, \frac{5}{2}\right).$$$
Answer
The new point is $$$\left(\frac{3}{2}, \frac{5}{2}\right) = \left(1.5, 2.5\right)$$$A.