Plane trigonometry, by S.L. Loney.

414

TRIGONOMETRY.

Ex. 1. Sum to n terms the series
3 32 33
sins 0 3 sins 30 +32 sin3 ~ q+...
Since always sin 30 = 3 sin 0 - 4 sin3 0, we have
0 1 O _
sin3 = (3sin -si - n 0),
i32 3_1-4L 323
00 0 1r 0.0]
3. sin3-.3.[3sm -smin = [ 8in2 -3sin 3,
32 sin=3 s 33 sn- 3in ],..............................................................................
3n-1 sins = [3n sin - 3n-' sin I].
Hence, by addition, the required sum
3 sin - sin 0].
Also the sum to infinity
= [0- sin 0]. (Art. 228.)
Ex. 2. Sum the series
tan a+ 2 tan 2a + 22 tan 22a.......- + 21-l tan 2n-1 a.
We have easily
tan a=cot a- 2 cot 2a,
tan 2a=cot 2a - 2 cot 22a,
tan 22a= cot 22a - 2 cot 23a,
and tan 2n- a = cot 2n-1 a - 2 cot 2na.
By multiplying these rows in succession by 1, 2, 22,...... 2n-1 we have
tan a + 2 tan 2a + 22 tan 22a..... + 2n- tan 2n-la = cot a - 2n cot 2na,
the other terms all disappearing.
The required sum therefore = cot a - 2n cot 2na.
Ex. 3. Sum the series
tan a tan (a + p) + tan (a + 3) tan (a + 2f) + tan (a + 2p) tan (a + 33) +......
to n terms.