Plane trigonometry, by S.L. Loney.

j42

TRIGONOMETRY.

Hence
2n cosn 0 = 2 cos nO + n.2 cos (n - 2) 0
n(n- l)co)+
+ 1 2 cos-.os(n4)......
i.e. 2'-1 cos" 0 = cos 0n + I cos (n- 2) 0
n (n - l)
+ ( ) cos(n-4) +..... (2).
If n be odd, there are an even number of ternms on the
right-hand side of (1), so that the terms take together in
pairs and the last term contains cos 0.
If n be even, there are an odd number of terms on the
right-hand side of (1), so that after all the possible pairs
have been taken there is a term left not containing x.
This term will, when divided by 2, form the last term on
the right-hand of (2).
290.. Ex. 1. Expand cos8 0 in a series of cosines of multiples of 0.
We have (2 cos 0)8_=( x+
1 1 1 1
=x8 +8x6+ 28x4+56x2+70 +56.,+28. +8. +x8) + 8 (6+ )+28 (x4 + )+56 (2+ 1 70
-( 8+ X4+7O
=2. cos 80 + 8. 2 cos 60 + 28. 2 cos 40 + 56. 2 cos 20 + 70,
27 cos 0=cos 80 +8 cos 60+ 28 cos40 +56 cos 20 +35.
Ex. 2. Expand cos7 0 in a series of cosines of multiples of 0.
We have (2 os0)7= (x+ 1)
7+ 1 1 1
=x7+7 x5+21x+35x+35. -+21.-+7. +x Z3 x +7
= (7+-1) +7 X5+ +21 X 3) + 35 (x +
=2.cos 70+7. 2 cos5+21.2 os+ 3+35. 2cos 6,
26 Cos7 = cos 70 + 7 cos 50 + 21 cos 30 + 35 cos 0.