In S L Loney's Trigonometry book on page 78, article 82, the author combines 2r$\pi$+$\alpha$ and (2r+1)$\pi$-$\alpha$ and arrives at this equation n$\pi$+(-1)n$\alpha$, which is a general expression to include all angles which have the same sine.
Then plug in n=2r and n=2r+1 in to the last equation and gets the first two respectively.
I understand n=2r is an even number of revolutions and n=2r+1 is an odd number of revolutions around the circle.
What does even and odd number of revolutions have to do with this ? What does 'n' stand for in the last equation ?
If you're asking how the author arrived at: $n\pi+(-1)^n\alpha$ from
$(2r+1)\pi-\alpha$ and $2r\pi+\alpha$
note that $(-1)^{2r+1} = -1$ and $(-1)^{2r} = 1$ when r is an integer