Deriving 2D Coordinate Rotation Formula I'm trying to write out the steps in code for deriving the 2D coordinate rotation formula so I can understand it.
x = radius * cos(angle)
y = radius * sin(angle)
x1 = radius * cos(angle + -rotation)
y1 = radius * sin(angle +  rotation)

So,
x1 = radius * cos(angle) * cos(rotation) – radius * sin(angle) * sin(rotation)
y1 = radius * sin(angle) * cos(rotation) + radius * cos(angle) * sin(rotation)

Therefore,
x1 = cos(rotation) * x – sin(rotation) * y
y1 = cos(rotation) * y + sin(rotation) * x

I hate to post a question just asking "Is this right?", but is this right? In particular, I'm unsure if I'm correctly representing the w (as listed in the link) in the first x1 and y1 assignment, and it's expansion. And no, this isn't homework. Thanks.

Why don't you look at the rotation this way. What is the coordinates of the rotated $\hat{x}=(1,0)$ axis? From trigonometry you will find it as $X=(\cos\theta,\sin\theta)$. Now what is the coordinates of the rotated $\hat{y}=(0,1)$ axis? Similarly from trig you get $Y=(-\sin\theta,\cos\theta)$
A general vector $\vec{v}=(A,B)=A \hat{x}+B \hat{y}$ is rotated by rotating the unit vectors $V = A*(\cos\theta,\sin\theta) + B(-\sin\theta,\cos\theta)$ or re-arranged
$$ V = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} \begin{pmatrix} A \\ B \end{pmatrix} $$
So the rotation matrix is $${\rm Rot}(\theta)=\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$$
