I am currently working with vectors in $\mathbb{R}$^2. For my specific project, it is beneficial to identify a non-standard basis as "new standard basis" of $\mathbb{R}$^2. For example, instead of working with basis $B$ (Euclidean coordinates)
$$B = \left\lbrace \begin{pmatrix}1 \\ 1\end{pmatrix}, \begin{pmatrix}1 \\ \sqrt{3}\end{pmatrix} \right\rbrace,$$
I do my calculations with my "new standard basis" $B'$
$$B' = \left\lbrace \begin{pmatrix}1 \\ 0\end{pmatrix}, \begin{pmatrix}0 \\ 1\end{pmatrix} \right\rbrace.$$
Here, I identify $\begin{pmatrix}1 \\ 1\end{pmatrix}$ with $\begin{pmatrix}1 \\ 0\end{pmatrix}$ and $\begin{pmatrix}1 \\ \sqrt{3}\end{pmatrix}$ with $\begin{pmatrix}0 \\ 1\end{pmatrix}$. So, when working with $B'$, vector $\begin{pmatrix}a \\ b\end{pmatrix}$ corresponds to $\begin{pmatrix}a + b\\ a+\sqrt{3}b\end{pmatrix}$ using $B$.
Question: Working with $B'$, I calculate the angle between, e.g., $\begin{pmatrix}2 \\ 3\end{pmatrix}$ and $\begin{pmatrix}4 \\ 5\end{pmatrix}$ by $$\cos\alpha' = \frac{\begin{pmatrix}2 \\ 3\end{pmatrix}\begin{pmatrix}4 \\ 5\end{pmatrix}}{\left|\left|\begin{pmatrix}2 \\ 3\end{pmatrix}\right|\right| \left|\left|\begin{pmatrix}4 \\ 5\end{pmatrix}\right|\right|}.$$ I am looking for a formula that tells me how $\alpha'$ changes when switching the base back to $B$, i.e. a what is the angle $\alpha$ between $$\begin{pmatrix}1 & 1\\ 1 & \sqrt{3}\end{pmatrix} \begin{pmatrix}2 \\ 3\end{pmatrix} \qquad \text{and} \qquad \begin{pmatrix}1 & 1\\ 1 & \sqrt{3}\end{pmatrix} \begin{pmatrix}3 \\ 4\end{pmatrix}$$
Please note: I am looking for a formula to transform $\alpha'$ to $\alpha$, I want to avoid recalculation from scratch.