$\displaystyle 3u_x+4u_y-2u=1\Rightarrow\omega_{\xi}+k\omega=\varphi(\xi,\e ta)$

The book states, "If the $\displaystyle (\xi,\eta)$-axes are obtained from $\displaystyle (x,y)$-axes by rotating through an angle $\displaystyle \alpha$, then $\displaystyle (\xi,\eta)$ and $\displaystyle (x,y)$ are related by either of the pair of equations:

$\displaystyle \xi=x\cos{\alpha}+y\sin{\alpha}, \ \ x=\xi\cos{\alpha}-\eta\sin{\alpha},$

$\displaystyle \eta=-x\sin{\alpha}+y\cos{\alpha}, \ \ y=\xi\sin{\alpha}+\eta\sin{\alpha}.\mbox{"}$

How did the book obtain those equations?