the moon - Calculate Atmospheric Extinction Using Source Altitude Angle

You seem to be referring to the computation of $m$ in equation (15) for the extinction as the light from the source passes through the atmosphere. The author intends to say that $mapprox frac{1}{sin(alpha_s)}$, where $alpha_s$ is the altitude of the source (in radian), so restricted to the range $0$ to $ pi/2$. (Note this is not the inverse of the sine function as the author says but the reciprocal of the sine function) The altitude is the angle measured from the horizon to the source, which is $=pi/2-theta$ where $theta$ is as in figure 2. Note this blows up when $alpha_s=0$ ($theta=pi/2$) because in the simple approximation being used the path length through the atmosphere in infinite.

The approximation being used here is that of a flat Earth, and $m$ is the number of equivalent atmosphere heights that the light path travellels through