Answer
$(4.5\times10^{-4})^{\circ}$
Work Step by Step
The distance between the Earth and the Moon is: $ d= 3.82\times10^8\;m$
The laser beam forms a spot on the Moon with a diameter of
$D=3\;km$
Therefore, the radius is: $r=1.5\;km=1.5\times10^{3}\;m$
Now, the angular divergence of the beam is
$\theta=2\tan^{-1}\Big(\frac{r}{d}\Big)$
or, $\theta=2\tan^{-1}\Big(\frac{1.5\times10^{3}}{3.82\times10^{8}}\Big)$
or, $\theta=(4.5\times10^{-4})^{\circ}$
