Plane trigonometry, by S.L. Loney.

EXPANSIONS OF SIN nO AND COS nO.

357

EXAMPLES. LII.
1. sin 78= 7 sin - 56 sin3 + 112 sin5 - 64 sin7 8.
2. cos 7=64cos7 -112 cos5 + 56 cos3 - 7 cos.
3. sin 88= sin 8 [128 cos7 8 - 192 cos5 0 + 80 cos3 0 - 8 cos 0].
4. cos 80 = 1 - 32 sin2 8 + 160 sin4 8 - 256 sin6 0 + 128 sin8 0.
5. sin 9 sin 8 [256 cos8 0 - 448 cos6 0 + 240 cos4 8 - 40 cos2 0 +1].
6. Express cos 60 in terms of cos 0 only and verify for the cases
7r 7
3' 2
respectively.
7, Prove the algebraic identity
pn + q= (p + q)t - n (p + q)n-2pq + 1. (p +q),3-4p2 q2.
Deduce that
n (n- 3)
2 cos n0 = (2 cos) - n (2os)-2 + (2cos0)'-....
* # 300. Ex. Prove that the roots of the equation
8x3 - 4x2 - 4x + 1 = 0
7r 3r7 57
are cos o, cos-7 and cos -7
7r 37 57r 1
and hence that cos - + cos - — 2
7T 37 37r 57r 57r 7r 1
cos - cos -+ cos - cos - + cos-y cos -
7 37r 57r 1
and cos 7 cos 7 cos -=- 8'
On putting n=7 in equation (2) of Art. 294, we have
7 4 7.3.2
2 cos 70 = (2 cos 0)7 - 7 (2 cos 0)5 + — (2 os 3 2 3(2 os
i.e., on reduction,
cos 70 = 64 cos7 - 112 cos5 + 56 cos3 8 - 7 cos..........(1).
Now put cos 70= - 1, so that
7r 37r 57
0=, or or.......
7' 7' 7