Translated from
:
http://www.metro-hs.ac.jp/rs/sinohara/zahyou_rot/zahyou_rotate.htm
_
The rotation of the origin is
shown in the figure below. On the 2-dimensional coordinates, there is a point p(x, y), the length of the line op is r, and the angle between the line op and the positive direction of the x-axis is a. The line op is rotated counterclockwise by b degrees around the origin to reach p' (s,t) 
s = r cos(a + b) = r cos(a)cos(b) – r sin(a)sin(b) (1.1)
t = r sin(a + b) = r sin(a)cos(b) + r cos(a) sin(b) (1.2)
where x = r cos(a) , y = r sin(a )
into (1.1), (1.2) ,
s = x cos(b) – y sin(b) (1.3)
t = x sin(b) + y cos(b) (1.4)
Expressed in determinant as follows:
2. The rotation of the coordinate system is
in the original coordinate system xoy, rotating theta degrees counterclockwise around the origin to become the coordinate system sot.
There is a certain point p, the coordinates in the original coordinate system are (x, y), and the new coordinates after rotation are (s, t). 
oa = y sin(theta) (2.1)
as = x cos(theta) (2.2)
Synthesizing (2.1), (2.2) 2
s = os = oa + as = x cos(theta) + y sin(theta)
t = ot = ay – ab = y cos(theta) – x sin(theta)
Expressed in determinant as follows:
