Rotation Calculator

The calculator will rotate the given point around another given point (counterclockwise or clockwise), with steps shown.

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Your Input

Rotate $$$\left(3, 7\right)$$$ by the angle $$$45^0$$$ 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, $$$x = 3$$$, $$$y = 7$$$, and $$$\theta = 45^0$$$.

Therefore, the new point is $$$\left(3 \cos{\left(45^0 \right)} - 7 \sin{\left(45^0 \right)}, 3 \sin{\left(45^0 \right)} + 7 \cos{\left(45^0 \right)}\right) = \left(- 2 \sqrt{2}, 5 \sqrt{2}\right).$$$

Answer

The new point is $$$\left(- 2 \sqrt{2}, 5 \sqrt{2}\right)\approx \left(-2.82842712474619, 7.071067811865475\right)$$$A.