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Hello Alexander.

Welcome aboard!

In many nineteenth century treatments of trigonometry you'll find that sin(1") factor inserted in various formulae. It is indeed related to the small angle approximation, but I would say that it was really their way in that era of saying "in radians". In modern mathematical notation, we would say:
sin(x) = x (for x small enough),
but they would say:
sin(x) = sin(1")*x.
The sine of one second of arc is extremely close to (2*pi)/(360*3600), differing at the twelfth decimal point (if I remember correctly), so it works well. You can easily convert to modern trigometric fashion by dropping sin(1") wherever you come across it in those older works.

-FER

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