When finding a coordinate using Euler form, why do we use: $$ \left\{ \begin{aligned} u(x,y) &= x\cos(a) - y\sin(a) \\ v(x,y) &= x\sin(a) + y\cos(a) \end{aligned} \right. $$ with $a$ being the angle.
I need to know why you switch this $\cos$ and $\sin$ around for $u$ and $v$ as well as why it's subtract for $u$ and add for $v$?
Think about polar coordinates. If $$ \left\{ \begin{aligned} x &= r\cos(\theta) \\ y &= r\sin(\theta) \end{aligned} \right. $$ for some $(r, \theta)$. The crucial observation is that in the new coordinate system rotated through angle $a$, the polar coordinates are $(r, \theta + a)$. In other words, the radial distance remains unchanged, and the angle coordinate shifts by $a$.
The new rectangular (cartesian) coordinates $(u, v)$ in terms of these new polar coordinates are $$ \left\{ \begin{aligned} u &= r\cos(\theta + a) \\ v &= r\sin(\theta + a) \end{aligned} \right. $$
Now, we apply the sum-of-angles identities to obtain the result: $$ \left\{ \begin{aligned} u &= r \, \bigl( \cos \theta \cos a - \sin \theta \sin a \bigr) = (r \cos \theta) \cos a - (r \sin \theta) \sin a = x \, \cos a - y \, \sin a\\ v &= r \, \bigl( \cos \theta \sin a + \sin \theta \cos a \bigr) = (r \cos \theta) \sin a + (r \sin \theta) \cos a = x \, \sin a + y \, \cos a \end{aligned} \right. $$