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same circle, what will be the number of sides, when the whole space intercepted between the two polygonal boundaries is an assigned part of either polygon? Ex. What is the figure when the exterior area is of the interior polygon?

(6) In a right-angled triangle, if the hypothenuse (c) be divided into segments (x), (y) by the line which bisects the right angle, and t the tangent of half the difference of the acute angles,

x y1t: 1-t.

(7) To divide a given angle into two others, whose sincs shall be in the ratio of m: n.

(8) A circle is inscribed in an equilateral triangle, an equilateral triangle in the circle, a circle again in the latter. triangle, and so on; if r, 71, 72, 73, &c., be the radii of the circles, shew that

r = r1 + r2+ry + &c.

(9) If a, b, c be the sides of a triangle, and p, q, r perpendiculars from a point within the triangle bisecting the sides, prove that

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(10) It is required to determine the continued product of n terms of the series

1

1

(sin cos) (sin cos) (sin cos

1

1

4

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(11) The circumference of the inner of two concentric circles, of which R and r are the radii, is divided into n equal parts; shew that the sum of the squares of the lines drawn from the points of division to any point in the circumference of the outer circle = n (R2 + r2).

(12) The ratio between the area of an equilateral and equiangular decagon described about a circle, and that of another within the same circle, is equal to

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(13) In a plane triangle ABC, having given the sum of the sides AC, CB, the perpendicular from the vertex C upon the base AB; and the difference of the segments of the base made by the perpendicular; find the sides of the triangle.

(14) The length of a road in which the ascent is 1 foot in 5, from the foot of a hill to the top is a mile and twothirds. What will be the length of a zigzag road, in which

the ascent is 1 foot in 12?

(15) At each end of a horizontal base measured in a known direction from the place of an observer, the angle which the distance of the other end and a certain object subtends is observed, as also the angle of elevation of the object at one end of the base. Find its height and bearing.

(16) A staff one foot long stands on the top of a tower 200 feet high. Shew that the angle which it subtends at a point in the horizontal plane 100 feet from the base of the tower is nearly 6'51".

(17) Find the degrees, minutes, and seconds, in the angle whose circular measure is 1; also the values of cos mπ and tan-(-1)m (m an integer).

(18) Two circles have a common radius (r) and a circle is described touching this radius and the two circles: prove that the radius of the circle which touches the three

=

3r

56*

(19) At noon, a column in the E.S.E. cast upon the ground a shadow, the extremity of which was in the direction N.E.: the angle of elevation of the column being a°, and the distance of the extremity of the shadow from the column (a) feet, determine the height of the column.

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(21) If A, B, C be the angles of a triangle,

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(22) In a regular polygon of n sides inscribed in a circle. whose radius is (r), if a be the distance of any point from the centre, and perpendiculars be drawn from this point upon all the sides of the polygon, the sum of the squares of the lines joining the feet of the adjacent perpendiculars is

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(23) Four objects situated at unequal but given distances in the same straight line, appear to a spectator in the same plane with them to be at equal distances from each other, it is required to determine his position.

(24) If a', b' be the segments of the hypothenuse made by a line bisecting the right angle, then

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(25) A boy flying a kite at noon, when the wind was blowing a from the south, and the angular distance of the kite's shadow from the north was ẞ, the wind suddenly changed to a from the south, and the shadow to ẞ, from the north, and the kite was raised as much above 45° as it had before been below that elevation. Shew that 0° being the angular elevation of the sun, and 45° — go that of the kite at first,

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INDEX.

Abbreviation of trigonometrical quantities.

8.

Angle, circular measure of the, 3; divi-
sions of the, 1.

Angles, relation between the angles and
sides of a triangle, 52; to find the sine
and cosine of the sum and difference of
two, 25.
Angular ineasure, described, 2; examples,
3; unit of, 2.

Arc, series for sine and cosine in terms
of the, 109; remarks on the object and
use of, 110; summation of the series,
114.

Area of polygons, to find the, 80.
Area of a triangle, to find the, 56; am-

biguous case, 58; to find the area of a
triangle in terms of the radius of the
inscribed circle, 59; to find the area of
a triangle in terms of the radius of the
circumscribing circle, 60; to find the
radius of the circumscribing circle in
terms of the sides, 60; when two sides
and the included angle are given, 56.
Area of a triangle, examples, 60.

Centisimal division, 1.
Circle, ratio of circumference to diameter,
1; subdivisions of the, 2; examples, 2;
to find the radius of a circle, described
about a regular polygon of n sides, 80;
to find the area of a polygon of n sides
inscribed in a circle, 81; examples, 83.
Circumference of a circle, ratio of, to the
diameter, 1.

Cosecart, definition of, 8; ratio of, 8;
value of, 300, 450, and 600, 14, 15.
Cosine defined, 8.

Cosine of 150, 180, 360, 540, 749,
values of, 14, 34, 35; examples, 36; of
300, 450, 600, value of, 14; A, value
of, 55; observations on, 55; formulæ
expressing the sin. cos. and tan. of the
multiple n 6, 108.
Cosine, ratio of, 8.

Cosine and sine, series for, in the terms

of the arc, 109; remarks on the object
and use of, 110; summation of the
series, 114.

Cosine and sine of A B, to find the ex-
pressions for. 26.

Cosine, to find the, and sine, of the sum
and difference of two angles, 25.
Cosine 3 A, to find the expressions for,

28.

Cosine, to find, of an angle of a plane
triangle in terms of the sides, 52; when
the triangle is obtuse-angled, 52; when
the triangle is acute-angled, 53.
Cotangent, defined, 8; ratio of, 8.
Cotangent of 300, 450, 600, value of, 14,
15; (A + B) to express in terms of
cot. A and cot. B, 31.

Definitions, 8.

Demoivre's theorem explained, 104.
Diameter of a circle, ratio of, to the cir-
cumference, 1.

Distances, trigonometrical, calculation of:
formulæ and examples, 96,

English angular measure, described, 1;
method of reducing into French, 1.
Euler's series for sine and cos, in terms
of the arc explained, 109; remarks on
the object and use of, 110; sumination,
114.

Examples, miscellaneous, 116.

Formulæ expressing the sine, cosine, and
tangent of the multiple n 6, 108.
Formulæ of important equations to the
triangle, 55.

Formulæ of verification, 33.

French angular measures described, 1;
method of reducing to English, 1.
Functions, trigonometrical, described, 33;
tracing the values of, 12; results, 15;
examples, 15.

Heights, trigonometrical, calculation of:
formulæ and examples, 96.

Inverse trigonometrical functions de-
scribed, 32.

Miscellaneous examples, 116.
Measure, angular, examples of, 3; English
angular, 1; French angular, 1; method
of reducing one into the other, 1; unit
of angular, 2.

Measure, circular, of the angle, 3.

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