Plane trigonometry, by S.L. Loney.

356

TRIGONOMETRY.

and
n ra2 /n2 (n2 _ 92)
(- C1)2o Cos 2 1 ~cos2 + ( 4 cos40......
+ (-1)2 2n-1 (co 0)...... (2).
In these equations change 0 into - 0, and therefore
2
cos 0 into sin 0.
Then sin nO will become
sin(2 - n), i.e. (- 1) sin nO,
and cos nO will become
cos ( - n0) i.e. (-1)2 cos n.
On making these substitutions we have, if n be even,
sinn. n(n 2-) n2) n(n2- 22)(n2 _42) i
co = s=nsn O sin 6 + sin 5...
cos 0 i - 5
+ (- 1)2+ (2 sin )n-1......(3),
and
2 n2 (n- 2) in 0
cos nO = 1 sin -- ~ sin4 0
12 4
+...... (-1)2 2n-sin......(4).
**299. Equations (1) and (2) of Art. 297 and equations (1) and (2) of Art. 298 give the expansions of sin nO
and cos nO in ascending powers of cos 0 for the cases when n
is even or odd. Equations (3) and (4) of the same two
articles give the expansions of the same two quantities in
terms of sin 0.