# Rotate $\left(\frac{5 \sqrt{2}}{2}, \frac{\sqrt{2}}{2} + \frac{\sqrt{6}}{2}\right)$ by $45^{\circ}$ counterclockwise around $\left(0, 0\right)$

The calculator will rotate the point $\left(\frac{5 \sqrt{2}}{2}, \frac{\sqrt{2}}{2} + \frac{\sqrt{6}}{2}\right)$ by the angle $45^{\circ}$ counterclockwise around the point $\left(0, 0\right)$, with steps shown.
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The origin is the point $\left(0, 0\right)$.

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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 $\left(\frac{5 \sqrt{2}}{2} \cos{\left(45^{\circ} \right)} - \left(\frac{\sqrt{2}}{2} + \frac{\sqrt{6}}{2}\right) \sin{\left(45^{\circ} \right)}, \frac{5 \sqrt{2}}{2} \sin{\left(45^{\circ} \right)} + \left(\frac{\sqrt{2}}{2} + \frac{\sqrt{6}}{2}\right) \cos{\left(45^{\circ} \right)}\right) = \left(2 - \frac{\sqrt{3}}{2}, \frac{\sqrt{3} + 6}{2}\right).$

The new point is $\left(2 - \frac{\sqrt{3}}{2}, \frac{\sqrt{3} + 6}{2}\right)\approx \left(1.133974596215561, 3.866025403784439\right).$A