How to derive rotation of a point?

Question

Suppose a point has coordinates (x,y) and we rotate it through an angle $\alpha$, now it has new coordinates (X,Y). I know the values of the new coordinates are:-

$$X =x\cos\alpha-y\sin\alpha$$ $$Y =x\sin\alpha+y\cos\alpha$$

I was trying to find the derivation of this formula but none of my books have it and am not able to find it on the internet or able to derive it myself, does anyone know its derivation, even a hint will be appreciated.

Answer 1

Try reading this, it provides some nice illustration you might like. Basically, you express $x,y$ in their polar form then uses the formula $$ \cos(a+b)=\cos(a)\cos(b)-\sin(a)\sin(b) \\ \sin(a+b)=\sin(a)\cos(b)+\cos(a)\sin(b) $$

Answer 2

It gets easier to understand if you know the polar form of complex numbears, $r\exp(i\alpha)$, where $r$ is the absolute value.

This can be written as $r\cos(\alpha) + i r\sin(\alpha)$ (you can derive this from Eulers formulas if you want). So when multiplying two such numbers, their $\alpha$s are added to each other, and the $r$s are multiplied. A rotation is like a complex number with $r=1$, so it does not change the length.