Plane trigonometry, by S.L. Loney.

EXPANSIONS. 419
359. Ex. If sin x = n sin (a + x), expand x in a
series of ascending powers of n, where n is less than unity.
Since
sin x = n sin (a + x) = n (sin a cos x + cos a sin x),
n sin a.. tan x =
- n cos a
exi - e-xi ni sin a
exi + 'e-i -- ncos a'
ei 1 -n cos a + ni sin a 1 - ne-ai
"e-i - nz cos a - ni sin a 1- nei '.. 2xi = log (1 - ne-i) - log (1 - nei)
=- ne-a - - -ea 3e......
2 3
1 e 1
+ neai + n2e2i + n3e3ai......
2 3
= n i (e2i - e-2ai)
1
+ I n3 (e3ai - e-3)i)......ad inf.
3 3. n. a In 2i n sin 2a + t 1 n. 2i sin 3 +.(1).
2.=nsinn1a+a3s ina+.......
In this equation we have assumed x to lie between
- and +; if it do not, then, instead of 2xi, we should
read 2k7ri + 2xi; the left hand of equation (1) would then
be x + k7-, and we must choose k so that x + krw shall lie
between - - and +.
2 2
As before the expansions are legitimate if n be < unity.
27-2