Plane trigonometry, by S.L. Loney.

284

TRIGONOMETRY.

In a similar manner we can obtain the sum of the cubes of the sines
of a series of angles in A.P.
Cor. Since
2sin2 a=l - cos2a, and 2 cos2 a = 1 + cos 2a,
we can obtain the sum of the squares.
Since again 8 sin4 a = 2 [1 - cos 2a]2
=2- 4 cos 2a + 2 cos2 2a = 3 - 4 cos 2a + cos 4a,
we can obtain the sum of the 4th powers of the sines. Similarly for the
cosines.
Ex. 3. Sum to n terms the series
cos a sin 3 + cos 3a sin 2i3 + cos 5a sin 33 +... to n terms.
Let S denote the series.
Then
2S= {sin (a+ /) - sin (a - )} + {sin (3a + 2/) - sin (3a - 2p)}
+ {sin (5a+ 3p) -sin (5a - 3) +..
= {sin (a + /) + sin (3a + 23) + sin (5a + 33) +...}
-{ sin (a-) + sin (3a-2 2) + sin (5a-3j3) +...}
sin ((a+/ )+ n-1 (2a + 13)} sin n2a
2a+3p
sin 2
sin n(a_3)+21(2aI-3) sinn2a -
2a - - by Art. 241,
sin -
n+1 ). n(2a+~3)
sin na + 1 P si n 22a __. 2a+ 3
sn 2
sin {na -- - sin n(2. 2a-i/
smin 2