Plane trigonometry, by S.L. Loney.

SIMPLE TRIGONOMETRICAL SERIES.

283

and sina+Sina a+ ) +sin a + +... to n terms=0.
244. Ex. 1. Find the sum of
sin a - sin (a + 3) + sin (a + 23) -... to n terms.
We have, by Art. 73,
sin (a + 3 + 7r)= - sin (a + 3),
sin (a + 2/ + 2-r) = sin (a + 23),
sin (a + 3p + 37r) =- sin (a + 33),
Hence the series
=sin a + sin (a + + 7r) + sin {a +2 ( 3+ r)
+sin {a+3 (3+7r)} +...
sin -2 (/37r)j sin (sn n )
= 2 -- -— i-, by Art. 241,
sin 2+
2 2
sin 1a+ 21(3+w1 sin 2m +v
Cos -
2
Ex. 2. Find the sum of the series
cos3 a + cos3 2a + cos3 3a +..... to n terms.
By Art. 107, we have
cos 3a =4 cos3 a - 3 cos a,
so that 4 cos3 a= 3 cos a + cos 3a.
So 4 cos3 2a = 3 cos 2a + cos 6a,
4 cos3 3a = 3 cos 3a + cos 9a,....................................
Hence, if S be the given series, we have
4S = (3 cos a + cos 3a) + (3 cos 2a + cos 6a) + (3 cos 3a + cos 9a) +...
=3 (cos a cos 2a cos3a+...) (cos3a + cos 6a + cos 9a +...)
fO n - 1 na n-I n. 3a
cos {a+n2. al sin - cos t3a+ 2. 3a snin
— 3 2 1 2 2 2
a. 3a
sin sinsm
n+1. na 3(n+ 1).3na
cos - a sin cos 2 a sin -
-=3 -... +-,,.
a. 3a
sin smin
2 2