# Rotation Calculator

## Rotate a point around another point step by step

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

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The origin is the point $\left(0, 0\right)$.

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

Rotate $\left(3, 7\right)$ 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, $x = 3$, $y = 7$, and $\theta = 45^{\circ}$.

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

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