Past exam of the mathematical methods II course of the University of Oxford / 2019 / 2 / Solution

The solid angle is the intercepted surface area on a sphere of radius $R$, divided by $R^2$. Take the cone axis as the polar axis of spherical coordinates, and call its half-opening angle $\theta$. With polar integration variable $\vartheta$,For small $\theta$ measured in radians, the Taylor series of the cosine givesThis is the solid angle of a circular cone, measured in steradians.

The supplied angular diameter is twice the half-opening angle, soHence either apparent disc subtendsThe fraction is about $0.0005409\%$ of the entire celestial sphere. Using the angular diameter directly in $\pi {\theta }^2$ would give four times the correct small-angle answer.