|
Rotate $$$\left(\frac{5 \sqrt{2}}{2}, \frac{\sqrt{2}}{2} + \frac{\sqrt{6}}{2}\right)$$$ by $$$45^{\circ}$$$ counterclockwise around $$$\left(0, 0\right)$$$
Your Input
Rotate $$$\left(\frac{5 \sqrt{2}}{2}, \frac{\sqrt{2}}{2} + \frac{\sqrt{6}}{2}\right)$$$ 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, $$$x = \frac{5 \sqrt{2}}{2}$$$, $$$y = \frac{\sqrt{2}}{2} + \frac{\sqrt{6}}{2}$$$, and $$$\theta = 45^{\circ}$$$.
Therefore, the new point is $$$\left(\frac{5 \sqrt{2}}{2} \cos{\left(45^{\circ} \right)} - \left(\frac{\sqrt{2}}{2} + \frac{\sqrt{6}}{2}\right) \sin{\left(45^{\circ} \right)}, \frac{5 \sqrt{2}}{2} \sin{\left(45^{\circ} \right)} + \left(\frac{\sqrt{2}}{2} + \frac{\sqrt{6}}{2}\right) \cos{\left(45^{\circ} \right)}\right) = \left(2 - \frac{\sqrt{3}}{2}, \frac{\sqrt{3} + 6}{2}\right).$$$
Answer
The new point is $$$\left(2 - \frac{\sqrt{3}}{2}, \frac{\sqrt{3} + 6}{2}\right)\approx \left(1.133974596215561, 3.866025403784439\right).$$$A