What is the prerequisite trigonometry knowledge required to understand 2D vector rotation? I am trying to understand the why of $\,2$D vector rotation. I know that I can use the following matrix to rotate a vector:
$$
\begin{bmatrix}
\cos\theta & -\sin\theta \\
\sin\theta & \cos\theta \\
\end{bmatrix}
$$
I do not understand why this works; I know that trigonometry is involved (where my level of knowledge is basically to fallback to "SOH CAH TOA" to remember which function to use). I can identify the first column as being similar to how a point on a circle is calculated i.e. $$r\!\cdot\!\cos\theta\,,\;r\!\cdot\!\sin\theta$$
But as for the second column I am lost. I have been reading up on trigonometric addition which seems like the area I need to focus on, but I have not been able to map the formula to the matrix form.
 A: If I had to guess, I would say that the intuitions you need to apply are about vectors and matrices, not trigonometry. An important thing to know about matrix multiplication is that the columns of a matrix tell what happen to the basis vectors. In particular, a matrix will map the basis vector $\left(\matrix{1\cr 0 \cr 0\cr \vdots}\right)$ to a linear combination of basis vectors with coefficients specified by the first column of the matrix. So for example, let's look at the first column of your matrix: $\left(\matrix{\cos\theta \cr \sin\theta}\right)$. What this means is that the rotation matrix maps the basis vector $\left(\matrix{1\cr 0}\right)$ to the vector $\left(\matrix{1\cr 0}\right)\cos\theta + \left(\matrix{0\cr 1}\right)\sin\theta$. (Another way to write this would be $\hat{\bf x} \mapsto \cos\theta\, \hat{\bf x} + \sin\theta\, \hat{\bf y}$.) Draw a picture and see if it makes sense that the counter-clockwise rotation of $\left(\matrix{1\cr 0}\right)$ by angle $\theta$ should result in the vector sum $\left(\matrix{1\cr 0}\right)\cos\theta + \left(\matrix{0\cr 1}\right)\sin\theta$. Then try applying similar reasoning to the second column of your rotation matrix.
Edit: Now I see by rereading that the second column was your real question. Draw a picture to see what happens to the vector $\left(\matrix{0\cr 1}\right)$ under counter-clockwise rotation by $\theta$. Express the result as a vector sum of the form $\underline{\qquad}\left(\matrix{1\cr 0}\right) + \underline{\qquad}\left(\matrix{0\cr 1}\right)$. You should find that the coefficients are the second column of your matrix.
A: This matrix comes from two facts.  One is the polar representation of complex numbers.  This is typically taught in trig classes.  We have that:
$$e^{i \theta} = \cos \theta + i \sin \theta$$
Using polar representation of complex numbers it’s easy to see that multiplying by $e^{i \theta}$ geometrically is a rotation by $\theta$ radians.
The second fact we use is that each complex number can be thought of as a 2x2 matrix.  The matrix you have is the matrix representation of $e^{i \theta}$.
https://www.nagwa.com/en/explainers/152196980513/
