Rotation in Coordinate Geometry

Let

P=(x,y)

Write the vector OP as

\begin{pmatrix} x\\ y \end{pmatrix}

After a counterclockwise rotation by θ, the horizontal unit vector

\begin{pmatrix} 1\\ 0 \end{pmatrix}

becomes

\begin{pmatrix} \cos\theta\\ \sin\theta \end{pmatrix}

while the vertical unit vector

\begin{pmatrix} 0\\ 1 \end{pmatrix}

becomes

\begin{pmatrix} -\sin\theta\\ \cos\theta \end{pmatrix}

Since

\begin{pmatrix} x\\ y \end{pmatrix} = x \begin{pmatrix} 1\\ 0 \end{pmatrix} + y \begin{pmatrix} 0\\ 1 \end{pmatrix}

its rotated image is

x \begin{pmatrix} \cos\theta\\ \sin\theta \end{pmatrix} + y \begin{pmatrix} -\sin\theta\\ \cos\theta \end{pmatrix}

Therefore

x'=x\cos\theta-y\sin\theta

y'=x\sin\theta+y\cos\theta