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LX. Pages 188, 189. (1) 34° 19' 31.8". (2) 1498.2 ft. (3) 45° 36' 56". (4) 5293-4 ft., 6982.3 ft. (5) 576.2 chains. (6) 4729 chains. (7) 3666'8 feet. (8) 42° 15', 11444 chains.

LXI. Page 190.

(1) 3842.9 ft. (2) 28174 ft. (3) 115.3 ft. (4) 285.6 ft. (5) 58° 17' 24", 31° 42' 36". (6) 656.1 chains, 41° 17' 12". (7) 81 ft.

(8) 1942 ft. (9) 646-7 miles. (10) 1000 ft.

LXVI. Pages 208, 209. (1) 41° 16' 51.5". (2) 73° 32' 12", 62° 46' 18". (3) 29° 17' 16", 31° 55' 31". (4) 64° 31' 58". (5) 730, 23' 54'4". (6) 41° 24' 34:6". (7) 82° 49' 9". (8) 75°, 60°, 45°. (9) 135°, 30°, 150,

LXVII. Page 211. (1) 313.46 yds.

(2) 28.87 inches, 31:43 inches, (3) 1192.55 yds. (4) 22.415 ft. (5) 24.995 = 25 ft. nearly, 17-559 ft., 65° 59' 42".

LXVIII. Pages 213, 214. (1) 108° 36' 30", 31° 23' 30". (2) 93° 11' 49", 36° 48' 11". (3) 57" 27' 25'4", 62° 32' 34:6". (4) 64° 26' 47", 370, 7' 13". (5) 72° 12' 59". (6) 20-5 chains. (7) 122.7. (8) 71° 13' 50", 32° 16' 10".

LXIX. Pages 218, 219. (1) Arvi°18'21", C=88° 41'39"; or A=128° 41'39", C=11° 18'21". (2) B=70° 0'56", C= 59° 59' 4"; or, B=1099 59' 4", C=20° 0' 56". (3) B=38° 38' 24", C=91° 21' 36', c=155.3. (4) 61° 16' 10". (5) A=72° 4' 48", B=41° 56' 12"; or, A = 107° 55' 12',

B=6° 5' 48", b=17:56. (6) B is ambiguous; 60-3893 ft.

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LXX. Page 220. The angles are given correct to the nearest second. (1) 28° 35'39". (2) 104° 44' 39". (3) 32° 20'48". (4) 43° 40'. (5) 128° 23' 13". (6) 106531 ft. (7) 3437.6 yds. (8) 1728.2 chains. (9) 25376 yards. (10) A=66°27'48",B=12°55'12". (11) A=92°12'53", B=35° 37' 7". (12) B=29°1'40";C=74° 55'50”. (13) B=70° 35'24"; or, 109° 24'36". (14) B=51°56'17";or, 128°3'43". (15) B=62° 6'10";or, 117°53'50". (16) Very nearly 90°. (17) 1319-6 yds.

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LXXII. Page 229. (15) To find the point E in an unlimited straight line CE at which a finite straight line AB subtends the greatest angle, a circle must be described passing through A and B, and touching the line CE in the point E. In (15) the centre of the circle lies vertically above E, and in the horizontal line through the middle point of AB.

LXXIII, Page 239. (1) (i) 10 sq. ft. (ii) 43•3 sq. in. (iii) 148.13 sq. yds.

(iv) 84 sq. chains=8.4 acres (v) 100 sq. ft.

(vi) 151872 sq. yds. (2) 4, 103, 12, 14 ft.

(4) 8j ft.

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LXXVI. Pages 253–256. (10) - (x+y+z) (y +2-x)(2+2-y) (x+y-2).

4r + (11) (i) is {6n1 - 2+(-1)"}. (ii)

2(

mn)
(iii) =
a-ß

B n
+cot-1 {tan

n a - m
2rT + TT
(iv)
2(m+n)

(v) sin 20 (3 cos 20 – 1) (2 cos 20 +1)=0.

50 30 (vi) sin 40.cos sin

=0.

(vii) sin 80. xin 40=0.

2 (21) 223:17.

{tan at m sin B +n sin a

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2

APPENDIX.

PART I.

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(1) Simplify the formulæ

32 +02-aa
COS A =

a)
cos! A =
2bc

bc in the case of an equilateral triangle.

(2) The sides of a triangle are as 2:16:1+13, find the angles.

(3) The sides of a triangle are as 4, 222, 2(13-1), find the angles.

(4) Given C=120°, c=719, a=2, find b.
(5) Given A=60°, b=4/7, c=6° 17, find a.
(6) Given A=45°, B=60o and a=2, find c.

(7) The sides of a triangle are as 7:8:13, find the greatest angle.

(8) The sides of a triangle are 1, 2, 77, find the greatest angle.

(9) The sides of a triangle are as a : 6:1(a2 + ab ), find the greatest angle.

(10) When a :b :c as 3 : 4:5, find the greatest and least angles; given cos 36° 52' =·8.

(11) If a=5 miles, b=6 miles, c=10 miles, find the greatest angle. [cos 49° 33'='65.]

(13)

C=

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(12) If a=4, b=5, c=8, find C; given that cos 54°54' =-575.

The sides of a triangle are 7, 8, 13, find the greatest angle.

(14) Given C=18°, a=J5+ 1, c=1/5 - 1, find the other angles.

(15) If b=3, C=120°, c=13, find a and the sines of the other angles.

(16) Given A=105°, B=45°, c=12, solve the triangle.
(17) Given B=750, C=30°, c=8, solve the triangle.
(18) Given B=45°, c=275, b=1/50, solve the triangle.

(19) Given B=30°, c=150, b=50/3, show that of the two triangles which satisfy the data one will be isosceles and the other right-angled. Find the third side in the greatest of these triangles.

(20) Is the solution ambiguous when B=30°, c=150, b=75 ?

(21) If the angles adjacent to the base of a triangle are 2210 and 1122°, show that the perpendicular altitude will be half the base. (22) If a=2, b=4-2/3, c= /6(1/3-1), solve the triangle.

2.13 (23) If A=9°, B=45°, b=16, find c. (24) Given B=15°, b=13–1,c=/3+1, solve the triangle.

(25) Given sin B="25, a=5, b=2.5, find A. Draw a figure to explain the result.

(26) Given C=15°, c=4, a=4+ 48, solve the triangle.

(27) Two sides of a triangle are 3 J6 yards and 3 (W3+1) yards, and the included angle 45°, solve the triangle.

(28) If C=30°, b=100, c=45, is the triangle ambiguous ?
(29) Prove that if A=45o and B=60° then 2c=a(1+13).

(30) The cosines of two of the angles of a triangle are 1 and 3, find the ratio of the sides.

ANSWERS TO APPENDIX.

(1) cos A = 1, cos4=* 3. (2) 45°, 60°, 75o. (3) 130°, 30°, 15o. (4) 3. (5) 14.

(6) 1+ 3.

(7) 120°. (8) 1200.

(9) 1200. (10) 90°, 36° 52'. (11) 130° 27'. (12) 125° 6'. (13) 120°. (14) A=54° or 126', B=108° or 36o. (15) a=1. (16) C=30°, a=V3+1, b=2. (17) A=750, a=b=V3+1. (18) C=60° or 120°.

(19) 100/3.

(20) no. (22) A=1050, C= 60°, B=15°.

(23) *V3(V5+1). (24) A =90° or 60°, C=75° or 105°, a=2v2 or v6. (25) 300

(26) A=45° or 135°, B= 30° or 120°, b=V2 (1+V3) or V6 (1+13). (27) 60°, 75°, 6 yds. (28) It is impossible. (30) 15:8 V3:4+5+6.

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or 1500.

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CAMBRIDGE : PRINTED BY C. J. CLAY, M.A. & son, AT THE UNIVERSITY PRESS.

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