Plane trigonometry, by S.L. Loney.

CHAPTER XXVIII.

SUMMATION OF SERIES. EXPANSIONS IN SERIES.
350. WE shall now apply the results of the preceding
chapters to the summation of some trigonometrical series.
The chief series may be divided into four classes;
(1) Those depending for their summation on a
Geometrical Progression ultimately,
(2) Those depending ultimately on the Binomial
Theorem,
(3) Those depending ultimately on the Exponential
Theorem, including, as sub-cases, the Sine and Cosine
Series,
and (4) Those depending ultimately on the Logarithmic
Series and, as a sub-case, Gregory's Series.
351. In Arts. 352 355 we shall sum one example of
each of these classes. It will generally be found more
convenient in summing one of these series involving sines
of multiple angles (such as sin a, sin 2a, sin 3a...) to also
sum at the same time the companion series involving the
cosines of the same multiple angles
(i.e. cos a, cos 2a, cos 3a...).
The method will be best seen by a careful study of the
following four articles.