31 sin z(2 cos z − 1) = 2 sin - cos' 2 28. Prove 2 29. Express log105.832, log10 √(35), and log10.3048 in terms of log10 2, logio 3, log10 7. 30. If the angle opposite the side a be 60°, and if b, c be the remaining sides of the triangle, prove that (a + b + c) (b + c − a) = 3 bc. Prove that 31. Assuming 22 to be the circular measure of two right angles, express in degrees the angle whose circular measure is . Find the number of degrees in an angle whose circular measure is }. 32. Show from the definitions of the trigonometrical function that sin2 A+ cot2 4 + cos2 A = cosec2 4. sec A + 1 33. Prove sin x(2 cos x + 1) = 2 cos sin 2 34. Find the logarithms of √(27) and .037 to the base √3. tan A+ sec A+ 1 = 35. If (sin A+ sin B + sin C) (sin A + sin B sin C) = 3 sin A sin B, and A+B+C 180°, prove that C = 60°. 36. Given A 18°, B = 144°, and b = 1; solve the triangle. = 37. Give the trigonometrical definition of an angle. What angle does the minute-hand of a clock describe between twelve o'clock and 20 minutes to four ? 38. Express the cosine and the tangent of an angle in terms of the sine. The angle A is greater than 90°, but less than 180°, and sin A = }; find cos A. 39. Find all the values of between 0 and 2 for which cose + cos 2 0 = 0. 40. If in a triangle a cos A = b cos B, the triangle will be either isosceles or right-angled. 41. The sides are 1 foot and √3 feet respectively, and the angle opposite to the shorter side is 30°; solve the triangle. 42. The sides of a triangle are 2, 3, 4; find the greatest angle, having given log 2 = .30103, log 3= .47712, log tan 52° 15' = .11110, log tan 52° 14' = .11083. 43. Distinguish between Euclid's definition of an angle and the trigonometrical definition. What angle does the minute-hand of a clock describe between half-past four and a quarter-past six ? 44. Express the sine and the cosine of an angle in terms of the tangent. The angle A is greater than 180°, but less than 270°, and tan A = }. Find sin A. 45. Prove (i.) 2 cot A sin 2 4 = 1+ cot2 4 A+B+C 90°, sin 2 A+ sin 2 B + sin 2 C = 4 cos A cos B cos C. 46. Find all the values of @ between 0 and 2 π, for which sin + sin 20 = 0. 47. If in a triangle b cos A = a cos B, show that the triangle is isosceles. 48. The sides are 1 foot and √2 feet respectively, and the angle opposite to the shorter side is 30°. Solve the triangle. (ii.) Show that if 49. Express in degrees, minutes, and seconds (1) the angle whose circular measure is; (2) the angle whose circular measure is 5. = If the angle subtended at the centre of a circle by the side of a regular heptagon be the unit of angular measurement, by what number is an angle of 45° represented? 50. Prove that (sin 30° + cos 30°) (sin 120° + cos 120°) = sin 30°. 51. Prove the formulæ : cos2 (a + B) sin2 a = cos 8 cos (2 a + B); (1) (2) 52. Find solutions of the equations : (1) 5 tan2x sec2 x = :11; 53. Two sides of between them is 30°. 54. Given that (2) sin 50 - sin 3 0 = √2 ⋅ cos 4 0. · a triangle are 10 feet and 15 feet in length, and the angle What is its area? sin 40° 29′ = 0.64922, sin 40° 30′ = 0.64944; find the angle whose sine is 0.64930. 55. Express in circular measure (1) 10'; (2) of a right angle. If the angle subtended at the centre of a circle by the side of a regular pentagon be the unit of angular measurement, by what number is a right angle represented? 56. If sec a = 7, find tan a and cosec a. 57. Prove the formulæ : (1) cos2 (a B) sin2 (a + B) = cos 2 a cos 2 B ; (2) 1+tan a tan a = sec a. 7; 58. Find solutions of the equations: (1) 5 tan2x + sec2 x = (2) cos 50+ cos 30= √2. cos 4 0. 59. The lengths of the sides of a triangle are 3 feet, 5 feet, and 6 feet. What is its area? 60. Given that sin 38° 25′ = 0.62137; sin 38° 26′ = 0.62160; find the angle whose sine is (0.62150). 61. Which is greater, 76° or 1.2¢? 62. Determine geometrically cos 30° and cos 45°. If sin A be the arithmetic mean between sin B and cos B, then cos 2 A= cos2 (B + 45°). 63. Establish the following relations: (1) (2) (3) 64. Express log10 log10 7. = = cosec 2 A; tan2 A sin2 A tan2 A sin2 A; 2 cos(x + y). (28), log10 3.888, log10.1742 in terms of log10 3, log10 5, 65. Prove that sin (A + B) = sin A cos B + cos A sin B, and deduce the expression for cos (A + B). Show that sin A cos (B+ C) – sin B cos (A + C) = sin (A – B) cos C. (3) sin 66. One side of a triangular lawn is 102 feet long, its inclinations to the other sides being 70° 30', 78° 10', respectively. Determine the other sides and the area. log sin 70° 30′ = 9.974, log 102 = 2.009, log sin 78° 10′ = 9.990, log 185 = 2.267, log sin 31° 20' 9.716, log 192 = 2.283, log 2.301, log 9234 = 3.965. = 67. Which is greater, 126° or the angle whose circular measure is 2.3 ? 68. Establish the following relations: (1) (2) (3) cos (x-3y) 2 π cot2 Acos2 A tan A+ cot 2 A cos (3 x − y) sin 2x + sin 2 y + sin 2✓cosec2 A 4π 69. Given log10 2.3010300, log10 9.9542425; find without using tables, log105, log10 6, log10.0216, and log10 V (.375). 70. Prove that sin 30° + sin 120° = √2 cos 15°. 71. Establish the identities: (1) 1+ cos A + sin A = √2 (1 + cos A) (1 + sin A) ; cosec2 A (2) cosec 2 A = sin = cot2 A cos2 A; G = 4 sin 6 п 2 sin (x − y). sin 37 sin 57 п 72. The sides of a triangular lawn are 102, 185, and 192 feet in length, the smallest angle being approximately 31° 20'. Find its other angles and its area. log sin 31° 20' = 9.716, == log 102 2.009, log 1922.283, log 2.301, log 9234 = 3.965. log sin 70° 30' = 9.974, log sin 78° 10' = 9.990, 73. If the circumference of a circle be divided into five parts in arithmetical progression, the greatest part being six times the least, express in radians the angle each subtends at the centre. 74. Define the sine of an angle, wording your definition so as to include angles of any magnitude. Prove that sin (90° + A) = cos A, and and by means of these deduce the formulæ (1) cot2 A = cosec2 A − 1 ; (2) cot A+ cot2 A = cosec1 A - cosec2 A. 77. If sin B be the cos 2 B 2 cos2 (A +45°). = sin A, cos (180° + A) = cos A. Verify (2) when A = 30°. 76. Evaluate to 4 significant figures by the aid of the table of logarithms × V.(008931). 7.891 geometric mean between sin A and cos A, then 78. The lengths of two of the sides of a triangle are 1 foot and √2 feet respectively, the angle opposite the shorter side is 30°. Prove that there are two triangles which satisfy these conditions; find their angles, and show that their areas are in the ratio √3 +1: √3 − 1. 79. If the circumference of a circle be divided into six parts in arithmetical progression, the greatest being six times the least, express in radians the angle each subtends at the centre. 6.12 .4131 80. Define the tangent of an angle, wording your definition so as to include angles of any magnitude. == Prove that tan (90° + A) cot A, and by means of this formula deduce the formula tan (180° + A) = tan A. 81. Compute by means of tables the value of × 54.17. 82. Prove that cos (A + B): = cos A cos B - sin A sin B, and deduce the expression for sin (A + B). Show that cos A cos (B+ C) - cos B cos (A + C′ ) = sin (A − B) sin C. 0827 (3) cos 2 4π 6π + cos + cos + 4 cos 7 7 84. Two adjacent sides of a parallelogram 5 in. and 3 in. long respectively, include an angle of 60°. Find the lengths of the two diagonals and the area of the figure. 86. Prove that 85. Investigate the following formulæ : (1) sin 3 A = 2 sec2 A (1 + 2 cos A) sin A; (2) sin (+8) — sin 0 = cos @ sin 8 (1 √2(1 + cos A)(1 − sin A) ; = = (1) sin 10° + sin 50° = sin 70°; (2) √3+tan 40° + tan 80° = √3 tan 40° tan 80°; sin A sin B cos C 87. Prove by means of the logarithmic table that = 1.846 nearly. 1 7374 = COS sin B tan 0 tan 8). 88. The length of one side of a triangle is 1006.62 feet and the adjacent angles are 44° and 70°. Solve the triangle, having given L sin 44° 9.8417713, L sin 66° 9.9607302, log 7654321 6.8839067, L sin 70 9.9729858, log 1006.62 3.0028656, log 103543 =5.0151212. sin A cos C COS A = +1=0. 89. Find the length of the arc of a circle whose radius is 8 feet which subtends at the centre an angle of 50°, having given π = 3.1416. 90. Prove that sin A = sin (A – 180°). Find the sines of 30° and 2010°. 91. Given that the integral part of (3.1622)100000 contains fifty thousand digits, find log10 31622 to five places of decimals. 92. Prove that (1) cos2 A + cos2 B-2 cos A cos B cos (A + B) = sin2 (A + B) ; 93. Prove that in any triangle a2 cos 2 B + b2 cos 2 A = a2 + b2 - 4 ab sin A sin B. log 2.3010300, log sin 15° 43′ = 9.4327777. |