information:extraction
Reflection

If (x, y) are the Cartesian coordinates of a point, then (− x, y) are the coordinates of its reflection across the second coordinate axis (the y-axis), as if that line were a mirror. Likewise, (x, − y) are the coordinates of its reflection across the first coordinate axis (the x-axis). In more generality, reflection across a line through the origin making an angle θ {\displaystyle \theta } with the x-axis, is equivalent to replacing every point with coordinates (x, y) by the point with coordinates (x ′, y ′), where

x ′ = x cos 2 θ + y sin 2 θ y ′ = x sin 2 θ − y cos 2 θ. {\displaystyle {\begin{aligned}x'&=x\cos 2\theta +y\sin 2\theta \\y'&=x\sin 2\theta -y\cos 2\theta.\end{aligned}}}

Thus: (x ′, y ′) = ((x cos 2 θ + y sin 2 θ), (x sin 2 θ − y cos 2 θ)). {\displaystyle (x',y')=((x\cos 2\theta +y\sin 2\theta \,),(x\sin 2\theta -y\cos 2\theta \,)).}