Plane trigonometry, by S.L. Loney.

SIN a AND COS a EXPANDED IN A SERIES. 329
In equation (1) make 0 indefinitely small, a remaining
constant and therefore n becoming indefinitely great.
sin
Then - is, in the limit, equal to unity and so is
every power of (-ai ). (Art. 263.)
Also cos 0 is, in the limit, equal to unity and so also is
every power of cos 0. (Art. 262.)
Hence (1) becomes
a2 a4 a6
cosa l-= 1 +.. adinf.
280. To expand sin a in terms of a.
As in Art. 274, we have
n(n- 1)(n-2)
sin n0 =c n cos -3 0 sin n0 +...
1. 2.3
As before put nO = a, and we have
a co~_ 0sin0 - I0 1 - - 2
sin a = cos2-l sin 0 cos-3 0 sin3s
+C78' 1. 1.2.3
+ cos- 1.2.3.4.5c sin 0...
1.2.3.4.5. sin 0\ a (a- 0)(a -20) s- ( sin 6\3
=acos ( -) 1.2.3....
As in the last article make 0 indefinitely small, keeping
a finite, and we have
a3 a5 a7
sin a=a —3 + - - + -... ad inf.
13 15 1