Plane trigonometry, by S.L. Loney.

442

TRIGONOMETRY.

372. Sinh 0 and cosh 0 in products.
By Art. 314 we have
sinh 0 = - i sin (0i) and cosh 0 = cos (0i).
Also the series of Arts. 369 and 370, being formed on
the Addition Theorem are, by Art. 310, true when for 0
we read 6i.
/ (t22\ / )122/ 3022.
and cosh = l- 72i2 )(1-72 ( 2-, 7r2... ad inf.
V 2 rr2 I \' 527r2)
=(1 + 2) (1+32 2 ) (.2~Tl* *3** (2).
The products (1) and (2) are convergent. For we know (C. Smith's
Algebra, Art. 333) that the infinite product II(1 +u,) is convergent if the
series Mu, be convergent.
In the case of (1), Su,
e9 /, 1 1 1 \
= 2 t1+ +3+2,+ 4'.. )
and the latter series is known to be convergent.
373. Sums of powers of the reciprocals of all
natural numbers.
From the results of Arts. 369 and 370 we can deduce
the sums of some interesting series.
From Arts. 369 and 280 we have
(1- 1- 22)2 (1- 322).....ad inf.
sin 0 02 94
=- 1-~I -. ad inf.
E, 3+ _