What does the $ar$ in $arcos(B)$ stand for? Regarding to the Cosine formula solving with all sides and no angles?
$arcos(B)$ in the formula given  where is it derived?

You probably started with $b^2=a^2+c^2-2ac\cos B$, rearranged that to $\cos B=\frac {a^2+c^2-b^2}{2ac}$ and then saw $B=\arccos(\frac {a^2+c^2-b^2}{2ac})$  It is the inverse function to $\cos$, sometimes written $\cos^{-1}$.  It says $B$ is the angle whose cosine is $\frac {a^2+c^2-b^2}{2ac}$
