63. Prove the formula

1830

sin (a + b) = 2 sin a sin (a - b) 4 sin a sin2

b 2

and explain fully its use in the construction of the Trigonometrical Canon.

64. The sides of a triangle are in arithmetical progression, and its area is to that of an equilateral triangle of the same perimeter as 3:5. Find the ratio of the sides, and the value of the largest angle.

65. Expand the nth power of sin A in terms of sines and cosines of multiples of A; and write down the last term with its proper sign when n is of the form 4m + 2.

66. Given tan 3A = n tan A. Find A in terms of n: find also the value of n that A may be 15o.

67. If R, r be the radii of the circumscribed and inscribed circles of a regular polygon of m sides, and R', 'the corresponding radii for a regular polygon of 2m sides and of the same perimeter as the former, then Rr' = R'2 and R + r = 2r'.

68. Find the sines of the sum and difference of two arcs in terms of the sines and cosines of the arcs themselves.

69. Given the sides of a plane triangle, find the cosine of an angle; investigate formula adapted to logarithmic computation for the solution of the triangle, and explain which of the methods is best in particular cases.

70. Given tan A, tan B . . . find tan (A + B + thence deduce tan 7A.

‚.), and

find 0, 4, and ; and sum the series

(sec) + (sec)+(sec)+(sec) +

22

23

73. In any polygon with n sides A,A,, AA,, tively represented by a,, a,.... prove that

respec

= 0.

74. Resolve - 1 into its simple and quadratic factors.

xm

75. If a and b be the sides of a plane triangle, A and B their opposite angles, then will

hyp. log b hyp. log a = cos 2A - cos 2B

76. Investigate an expression for the sine of an angle of 1831 a plane triangle in terms of the sides.

whether n be integral or fractional.

78. Express (sin A)4n+1 in terms of the sines of the multiple

arcs.

79. From a station B at the base of a mountain, its summit A is seen at an elevation of 60°; after walking one mile towards the summit up a plane making 30° with the horizon, to another station C, the angle BCA is observed to be 135°. Find the height of the mountain in yards.

80. An indefinite area is to be divided into similar and equal regular figures. Shew by what figures this can be done. Also, if three equal areas be divided into the same number of equal regular figures, which are respectively triangular, square, and hexagonal, shew that the sum of the lengths of the dividing lines in the cases of triangular, square, and hexagonal divisions, are to one another as 27: 16:12; the whole area to be divided being very great in comparison of one of the divisions.

81. If a be less than 45°, shew that

for the area of a triangle in terms of the sides.

84. A person standing at the edge of a river observes that the top of a tower on the edge of the opposite side subtends an angle of 55° with a horizontal line drawn from his eye; receding backwards 30 feet, he then finds it to subtend an angle of 48°. Determine the breadth of the river.

log sin 7° 9.08589, log sin 35° = 9.75859,

log sin 48° 9.87107,

=

log 1.0493.02089.

log 3 = 47712,

85. Determine the distance between two inaccessible objects, by observations made at two stations, the distance between which is known.

+ =

86. Solve the cubic x3 + qxr0 by trigonometrical formulæ, in the case in which the three roots are possible.

87. If a straight line bisect at right angles any side AB of a regular polygon of an odd number of sides, shew that it will pass through the point of intersection of the two sides of the polygon which are most remote from AB.

If 2n+1 be the number of sides, prove that the length of the part of the bisecting line within the polygon is equal to x cot

AB

2

π

2 (2n + 1)'

Prove

88. If the tangents of all arcs less than 45° be found, the tangents of all greater arcs can be found by addition. also that 4 sin (0 — a) sin (m✪ — a) cos (0 — m0)

= 1 + cos (20 — 2m0) — cos (20 — 2a)

=

cos (2m0 - 2a).

89. Resolve x + 1 into its factors, m being odd.

90. Resolve sin into its factors. Shew also that in the determination of an angle, which is nearly 90°, from its logarithmic sine, a small error in the logarithmic sine will produce a large error in the angle.

91. Find the sine of 36°. Prove the formula

sin A = sin (36° +A) + sin (72° — A) — sin (36°— A) — sin 72°, and mention its use.

92. Define an angle and the sine and cosine of an angle.

93. Determine the distance between two visible but inaccessible objects in the same plane with an observer.

94. If A, B, C be the angles of a plane triangle, a, b, c any points in the sides respectively opposite to them, prove that the lines joining A, a; B, b; C, c, respectively will intersect in a point if

95. When x is possible, shew that all the possible values of

(a + b√ − 1)2 + (a − b √ − 1)*

96. Expand the cosine of a multiple arc in terms of the powers of the cosine of the simple arc.

97. Explain the construction of a table of logarithmic sines and cosines.

501 +80 10

98. Prove that

is a close approximation to

240

the known numerical value of the semi-circumference of a circle

whose radius is 1.

E

1833

1834

99. Find the area of an equilateral and equiangular polygon of n sides, circumscribed about a circle whose radius is r. What is its ultimate value, when n is increased indefinitely?

100. If R, r, be the radii of circles circumscribed about and inscribed in the same plane triangle, prove that the distance of the centres of these circles R2 — 2Rr.

=

102. Define the sine and cosine of an angle; prove that

sin (A + B) = sin A. cos B + cos A. sin B ;

and write down a general formula for all angles whose cosine = cos A.

103. Express the cosine of half an angle of a triangle in terms of the sides; and explain in what cases the formula may be used with advantage in determining the angle when the sides are given.

104. Explain the use of subsidiary angles in adapting algebraical formulæ to numerical calculation.

Reduce the expression

Na-b √a + b

+

a+b a

b

to a form adapted to logarithmic calculation.

105. Find x from the equation

nx = (√T + x − 1) (VT − x + 1),

(√T