Plane trigonometry, by S.L. Loney.

450

TRIGONOMETRY.

[Exs. LXV.]

18 sin a - sin = (1_ (- 0 )( 0
sin a a/ 7r -a1 7r-+a
(1 27r- a ( 27r+a)
19. 2 cosh 0+2cosa
4 2 a [1 + 0 [2 02
4COS2L (a + r)2iL (-a-......
=4 cos2 2 II [ l+ r 2
2 (a +.r)
where r is any integer positive or negative.
20. Prove that
r=n-I 2
sinh nu=n sinh u I 1 +,
r sin2
and deduce the expression for sinh u in the form of an infinite product of
quadratic factors in u.
[Start with the result, when 0 is zero, of Ex. 1, Art. 368. In this
result put 0 equal to zero and divide.]
21. Prove that the value of the infinite product
(i12) (1i22)(1+)32......adinf.
is - sinh 7r.
7r
22. A semicircle is divided into m equal parts and a concentric and
similarly situated semicircle is divided into n equal parts. Every point
of section of one semicircle is joined to every point of section of the
other. Find the arithmetic mean of the squares of the joining lines and
prove that when m and n are indefinitely increased the result is
a2+ b2 -,where a and b are the radii of the semicircles.
22
aa
23. The radii of an infinite series of concentric circles are a, 2' 3..
From a point at a distance c (> a) from their common centre a tangent
is drawn to each circle. Prove that
sin 01 sin 02 sin 0......=. - Sin,
where, are the angles that the tangents subtend at the
mmwhere 0,...... are the angles that the tangents subtend at the
common centre.