Plane trigonometry, by S.L. Loney.

406

TRIGONOMETRY.

The sum to infinity is obtained by omitting the terms
containing cn and cn+], which become indefinitely small
when n is very great.
1 -c cos a
Hence C 1-c cos a + c
I- 22ccos a -+C2'
~and =~ cc sin a
1 - 2c cos a + c2
From the results for C and S it is now clear that the above series
might have been summed, without the use of imaginary quantities, by
multiplying both sides of (1) and (2) by the quantity 1-2ccosa+c2.
The coefficients of c2, c3......cn-l would then be found to vanish and the
values of C and S be easily obtained.
353. Ex. Sum the series
1 1.3. 1.3 5
- sin a + 2- sin 2a + sin 3a +.... ad inf.
2 2.4 2.4.6
1 1.3 1.3.5.,
Let S = sina+ sin2a s3a+...
sin a+ - 2 - 4. s s.i
2 2.42.4.
1 1 3 1.3.5
and C =1 + cos a + - cos 2a +2 cos 3a +....
2 2.4 2.4.6
Hence, multiplying the first by i and adding to the
second, we. have
1 e 1.3 1.3.5 3i
C +S i = 1 + I eai + I ' 3 e2ai + i. 3 * 5 eai +
0+Si=1++e + 24. e2 + 24..6 e
= (1 - e/i)-, if a: 2n7r,
by the Binomial Theorem. (Art. 273.). + Si= {1 - cos -isin a}-2
= 2 sina sin a-icos a)
= {2 sin cos (2- + sin - 2
={2s 2- f} O s( )+7r - s ai
= ^2 sin a cos c + i sin "