Plane trigonometry, by S.L. Loney.

440

TRIGONOMETRY.

370. To express cos 0 as a product of an infinite series
of factors.
In equation (4) of Art. 369 write for 0 the quantity
7r
+ 0, and the equation becomes
7r+29. 37r+20 5r-7+20
cos 0 = 2p-1 sin + 20 sin 3 + 2 57+2
2p 2p 2p
i (2p- 1)7r + 20 (1)
sin 2p...... 0.
The last factor. r 7-2f9. 7r-29
=sin r- 7- - = sin 2-
the last but one
=sin (2p-3)7r +2] = sin 37r-20
L 2p J '
and so on.
Hence taking the factors in pairs, as before, we have
[ 7rr20 Tr-0r. 37+20. 37r-20 -cos = 2P-1 sm — sin 2 — in +2 s in3-2..
2psi 2p 2p 2 p p J
27r 2 20291. 371- 2 291
= 2-1 [sin 2 - sin2 [sin2 - sin2...(2).
In (2) make 0 zero and we have
7r 37r 57r
1 = 2p-1. sin2.i2. sin2.......(3)
2p 2p 2p.
Dividing (2) by (3), we have. si2 2- 220 si 291
sin2 - sin 2-p sin2
cos _= [- 1 2p H1 2p 2p
sin2 - sin2 sin2 I
2p 2p 2p