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Rotate $$$\left(\frac{5 \sqrt{2}}{2}, - \frac{\sqrt{6}}{2} + \frac{\sqrt{2}}{2}\right)$$$ by $$$45^{\circ}$$$ counterclockwise around $$$\left(0, 0\right)$$$
Your Input
Rotate $$$\left(\frac{5 \sqrt{2}}{2}, - \frac{\sqrt{6}}{2} + \frac{\sqrt{2}}{2}\right)$$$ 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, $$$x = \frac{5 \sqrt{2}}{2}$$$, $$$y = - \frac{\sqrt{6}}{2} + \frac{\sqrt{2}}{2}$$$, and $$$\theta = 45^{\circ}$$$.
Therefore, the new point is $$$\left(\frac{5 \sqrt{2}}{2} \cos{\left(45^{\circ} \right)} - \left(- \frac{\sqrt{6}}{2} + \frac{\sqrt{2}}{2}\right) \sin{\left(45^{\circ} \right)}, \frac{5 \sqrt{2}}{2} \sin{\left(45^{\circ} \right)} + \left(- \frac{\sqrt{6}}{2} + \frac{\sqrt{2}}{2}\right) \cos{\left(45^{\circ} \right)}\right) = \left(\frac{\sqrt{3} + 4}{2}, 3 - \frac{\sqrt{3}}{2}\right).$$$
Answer
The new point is $$$\left(\frac{\sqrt{3} + 4}{2}, 3 - \frac{\sqrt{3}}{2}\right)\approx \left(2.866025403784439, 2.133974596215561\right).$$$A