Plane trigonometry, by S.L. Loney.

EXPANSION OF SINn 0. 345
292. Ex. 1. Expand sin6 0 in a series of cosines of multiples of 0.
We have 26i6sin6 = x — )
1 11
= x6-6X4+ 15X2-20+15. -6. 1 +,
X2 X4 X6
so that - 26 sin6 0= 6 +) - 6(+ + 15(+ 12 - 20
=2 cos 60 - 6.2 cos 40 +15.2 cos 20 - 20.
- 25 sin5 0= cos 60 - 6 cos 40 +15 cos 20 -10.
Ex. 2. Expand sin7 0 in a series of sines of multiples of 0.
We have 27i7 sin7 0= x -
-x7-7x5+21x- x 1 1 1 1
=;7 - 7x5 213 - 35x 35 1. +21 - -7.
X X~ X5 X,
=(X7 - )-I) (X5- + 21 (x3- 3) - 35 (x ).
- 27. i. sin7 0 =2i sin 7 - 7.2i sin 50 +21.2i sin 30 - 35.2i sin 0.
- 26 sin70= sin 70- 7 sin 50 + 21 sin 30 - 35 sin 0.
Ex. 3. Expand cos5 0 sin7 0 in a series of sines of multiples of 0.
We have
25 cos5 0 = (x + and 27 i7 sin7 0 = (x -.
Hence 212. i7. cos5 0 sin7 0 = (x - ( -
-x1 _ [5 Io - 2 +-6- ] -2+ x
/~ 1 \ / 1 \ f 1\ / 1\
= 12 ) 2 (Ox l —) 4 (8 - + 10 (6
+5 (X4 )-20 (X2 )
Hence, as before, we have
- 211 cos5 0 sin7 0 = sin 120 - 2 sin 100 - 4 sin 80 + 10 sin 60 + 5 sin 40
- 20 sin 20.