Plane trigonometry, by S.L. Loney.

SIN 0 IN FACTORS. 437
Resolution of sin 0 and cos 0 into factors.
369. To express sin 0 as a product of an infinite series
of factors.
We have sin 0 = 2 sin ~ cos 0
2 sin sin ( +.........(1).
2 22) ~~~~1.
Similarly in (1) changing 0 into 0 and 2 + 0 successively, we have
sin 2 = 2 sin sin + - = 2sin si +
22 Y2
and 2 r; 22 22 2
and sin + =2 sin + ). sin (+ +
= sin (22 2. sin ( + 22 22
Substituting these values in the right-hand side of (1)
we have, after rearranging,
0. 7r+O. 27r+. S3w+O
sin 0 = 23 sn 2sm 2- sin n 2 Sn 2...(2).
22 22 22 22
Applying once more the formula (1) to each of the
terms on the right hand of (2) and arranging, we have
0. 7r+. 27r+ 0. 3r+. 47r+0
sin 0 = 27 sin sin m sin sin sin
5.7r+O. 67r+O. 77r+O
sin sm s i......(3).
Continuing this process we have finally
sin 0 = 2-1 sin sin + sin +... sin - +
P P P............ (4),
where p is a power of 2.