Rotate $$$\left(\frac{7 \sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$$$ by $$$45^{\circ}$$$ counterclockwise around $$$\left(0, 0\right)$$$

The calculator will rotate the point $$$\left(\frac{7 \sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$$$ 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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Your Input

Rotate $$$\left(\frac{7 \sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$$$ 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, $$$x = \frac{7 \sqrt{2}}{2}$$$, $$$y = \frac{\sqrt{2}}{2}$$$, and $$$\theta = 45^{\circ}$$$.

Therefore, the new point is $$$\left(\frac{7 \sqrt{2}}{2} \cos{\left(45^{\circ} \right)} - \frac{\sqrt{2}}{2} \sin{\left(45^{\circ} \right)}, \frac{7 \sqrt{2}}{2} \sin{\left(45^{\circ} \right)} + \frac{\sqrt{2}}{2} \cos{\left(45^{\circ} \right)}\right) = \left(3, 4\right).$$$

Answer

The new point is $$$\left(3, 4\right)$$$A.