Plane trigonometry, by S.L. Loney.

478

TRIGONOMETRY. [Misc. Exs. LXIX.]

9. Prove that the sum to infinity of the series
1 sin3 1.3 sin5 0
sin6+- + +-+......
2 3 2.4 5
is 0, if 0 be acute, and, generally, is nr + (-1)1 0, where n is so chosen
71r
that nr + (-1) 0 lies between - - and +.
2i 2
10. If the arc of a circle of radius unity be divided into it equal arcs,
and right-angled isosceles triangles be described on the chords of these arcs
as hypothenuses and have their vertices outwards, prove that when n is
indefinitely increased the limit of the product of the distances of the
vertices from the centre is e2, where a is the angle subtended by the arc
at the centre.
11. The sides of a regular polygon of n sides, which is inscribed in a
circle, meet the tangent at any point P of the circle in A, B, C, D......
Prove that the product OA. OB. OC. OD......= an tan n or angtan2 nO,
according as n is odd or even, where a is the radius of the circle and 0 is
the angle which the line joining P to an angular point subtends at the
circumference.
12. A regular polygon of n sides is inscribed in a circle and from any
point in the circumference chords are drawn to the angular points; if
these chords be denoted by cl, c2,... cn, beginning with the chord drawn
to the nearest angular point and taking the rest in order, prove that the
quantity
C1C2+C2C3+ *... + lCn- Cn-C
is independent of the position of the point from which the chords are
drawn.
13. A series of radii divide the circumference of a circle into 2n equal
parts; prove that the product of the perpendiculars let fall from any
point of the circumference upon n successive radii is
2 sin nO,
where r is the radius of the circle and 0 is the angle between one of the
extreme of these radii and the radius to the given point.
14. If a regular polygon of n sides be inscribed in a circle, and 1 be
the length of the chord joining any fixed point on the circle to one of the
angular points of the polygon, prove that
12m
s12M = na2m 2 -- 2m Mn2'