Are there calculation rules for rotating points on the cartesian plane? (NOT POLAR)

Question

I asked this question earlier, but was not specific enough. Are there calculation rules for sets of points when rotating them on the cartesian, 2D plane? I.E., the rule for rotating the set, (x,y) 270° CCW, it becomes (y,-x). And there are also "rules" for other rotations of multiples of 90°. Are there "rules" for numbers between those multiples? Like a "rule" for rotating a point 23° CCW?

Answer 1

You can see the rotation of a point in the plane by a angle $\theta$ as multiplying by the matrix $ \begin{pmatrix} cos \theta & - sin \theta \\ sin \theta & cos \theta \end{pmatrix}$, i.e. $$ R_{\theta}(x,y) = \begin{pmatrix} cos \theta & - sin \theta \\ sin \theta & cos \theta \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} xcos \theta - y sin \theta \\ x sin \theta +y cos \theta \end{pmatrix}.$$ This is the mor general rule and you can recover the examples you give.

You can find more information in internet as "rotation matrix".