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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 (0,0)\left(0, 0\right).

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please contact us.

Your Input

Rotate (3,7)\left(3, 7\right) by the angle 45∘45^{\circ} counterclockwise around (0,0)\left(0, 0\right).

Solution

Rotation of a point (x,y)\left(x, y\right) around the origin by the angle θ\theta counterclockwise will give a new point (xcos⁡(θ)−ysin⁡(θ),xsin⁡(θ)+ycos⁡(θ))\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=3x = 3, y=7y = 7, and θ=45∘\theta = 45^{\circ}.

Therefore, the new point is (3cos⁡(45∘)−7sin⁡(45∘),3sin⁡(45∘)+7cos⁡(45∘))=(−22,52).\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).

Answer

The new point is (−22,52)≈(−2.82842712474619,7.071067811865475)\left(- 2 \sqrt{2}, 5 \sqrt{2}\right)\approx \left(-2.82842712474619, 7.071067811865475\right)A.