Trigonometry - Review Questions Answers Rev 0

MECHANICAL ENGINEERING COMPREHENSIVE EVALAUTION COURSE 1 TOPIC: PLANE AND SPHERICAL TRIGONOMETRY Instruction: Answer the following multiple choice questions. 18. Find the value of y in the given: y = (1 + cos 2θ) tan θ. 1. Sin (B – A) is equal to _______, when B = 270 degrees and A. sin θ C. sin 2θ* A is an acute angle. B. cos θ D. cos 2θ A. – cos A* C. cos A B. – sin A D. sin A 19. Find the value of A. 2 sin θ C. 2 tan θ* 2. If sec2 A is 5/2, the quantity 1 – sin2 A is equivalent to? B. 2 cos θ D. 2 cot θ A. 2.5 C. 1.5 B. 0.4* D. 0.6 20. Simplify the equation sin2 θ (1 + cot2 θ) A. 1* C. sin2 θ sec2 θ 3. (cos A)4 – (sin A)4 is equal to ______. B. sin2 θ D. sec2 θ A. cos 4A C. cos 2A* B. sin 2A D. sin 4A 21. Simplify the expression sec θ – (sec θ) sin2 θ A. cos2 θ C. sin2 θ B. cos θ* D. sin θ 4. Of what quadrant is A, if sec A is positive and csc A is negative? A. IV* C. III 22. Arc tan [2 cos (arc sin [(31/2) / 2]) is equal to B. II D. I A. C. 5.

6.

csc 520o is equal to A. cos 20o* B. csc 20o

D.

24. Solve for x in the given equation: Arc tan (2x) + arc tan (x) = π/4 A. 0.149 C. 0.421 B. 0.281* D. 0.316

Solve for θ in the following equation: Sin 2θ = cos θ A. 30o* C. 60o B. 45o D. 15o

8.

If sin 3A = cos 6B, then A. A + B = 90o B. A + 2B = 30o*

25. Solve for x in the equation: arc tan (x + 1) + arc tan (x – 1) = arc tan (12). A. 1.5 C. 1.20 B. 1.34* D. 1.25

C. A + B = 180o D. None of these

26. Solve for A for the given equation cos2 A = 1 – cos2 A. A. 45, 125, 225, 335 degrees C. 45, 135, 225, 315 degrees* B. 45, 125, 225, 315 degrees D. 45, 150, 220, 315 degrees

Solve for x, if tan 3x = 5 tan x. A. 20.705o* C. 35.705o B. 30.705o D. 15.705o

27. Evaluate the following:

10. If sin x cos x + sin 2x = 1, what are the values of x? A. 32.2o, 69.3o C. 20.90o, 69.1o* B. – 20.67o, 69.3o D. – 32.2, 69.3o

A. 1* B. 0

11. Solve for G is csc (11G – 16 degrees) = sec (5G + 26 degrees). A. 7 degrees C. 6 degrees B. 5 degrees* D. 4 degrees o

*

23. Evaluate arc cot [2cos (arc sin 0.5)] A. 30o* C. 60o o B. 45 D. 90o

C. tan 45o D. sin 20o

7.

9.

B.

Angles are measured from the positive horizontal axis, and the positive direction is counter clockwise. What are the values of sin B and cos B in the 4th quadrant? A. sin B > 0 and cos B < 0 C. sin B > 0 and cos B > 0 B. sin B < 0 and cos B < 0 D. sin B < 0 and cos B > 0*

C. 45.5 D. 10

28. Simplify the following: A. 0* B. sin A

o

12. Find the value of A between 270 and 360 if sin 2 A – sin A = 1. A. 300o C. 310o B. 320o D. 330o*

C. 1 D. cos A

29. Evaluate:

13. If cos 65o + cos 55o = cos θ, find the θ in radians. A. 0.765 C. 1.213 B. 0.087* D. 1.421

A. sin θ B. cos θ

C. tan θ D. cot θ*

14. Find the value of sin (arc cos 15/17 ). A. 8/11 C. 8/15 B. 8/19 D. 8/17*

30. Solve for the value of “A” when sin A = 3.5x and cos A = 5.5x. A. 32.47°* C. 34.12° B. 33.68° D. 35.21°

15. The sine of a certain angle is 0.6, calculate the cotangent of the angle. A. 4/3* C. 4/5 B. 5/4 D.3/4

31. If sin A = 2.511x, cos A = 3.06x and sin 2A = 3.939x, find the value of x? A. 0.265 C. 0.562 B. 0.256* D. 0.625

16. If A. 5o B. 6o*

32. If coversed sin θ = 0.134, find the value of θ. A. 30o C. 60o* B. 45o D. 90o

, determine the angle of A in degrees. C. 3o D. 7o

33. A man standing on a 48.5 meter building high, has an eyesight height of 1.5 m from the top of the building, took a depression reading from the top of another nearby building and nearest wall, which are 50° and 80° respectively. Find the height of the nearby building in

17. If tan x = 1/2, tan y = 1/3, what is the value of tan (x + y)? A. ½ C. 2 B. 1/6 D. 1*

1

MECHANICAL ENGINEERING COMPREHENSIVE EVALAUTION COURSE 1 TOPIC: PLANE AND SPHERICAL TRIGONOMETRY meters. The man is standing at the edge of the building and the tower. The vertical angle at point A is 30° and at point both buildings lie on the same horizontal plane. B is 40°. What is the height of the tower? A. 39.49* C. 30.74 A. 85.60 feet C. 110.29 feet B. 35.50 D. 42.55 B. 92.54 feet* D. 143.97 feet 34. Points A and B 1000 m apart are plotted on a straight highway running East and West. From A, the bearing of a tower C is 32° W of N and from B the bearing of C is 26° N of E. Approximate the shortest distance of tower C to the highway. A. 364 m C. 384 m B. 374 m* D. 394 m

45. A PLDT tower and a monument stand on a level plane. The angles of depression of the top and bottom of the monument viewed from the top of the PLDT tower at 13° and 35° respectively. The height of the tower is 50 m. Find the height of the monument. A. 29.13 m C. 32.12 m B. 30.11 m D. 33.51 m*

35. Two triangles have equal bases. The altitude of one triangle is 3 units more than its base and the altitude of the other triangle is 3 units less than its base. Find the altitudes, if the areas of the triangles differ by 21 square units. A. 6 and 12 C. 5 and 11 B. 3 and 9 D. 4 and 10*

46. If an equilateral triangle is circumscribed about a circle of radius 10 cm, determine the side of the triangle. A. 34.64 cm* C. 36.44 cm B. 64.12 cm D. 32.10 cm 47. The two legs of a triangle are 300 and 150 m each, respectively. The angle opposite the 150 m side is 26°. What is the third side? A. 197.49 m C. 341.78 m* B. 218.61 m D. 282.15 m

36. A ship started sailing S 42°35’ W at the rate of 5kph. After 2 hours, ship B started at the same port going N 46°20’W at the rate of 7 kph. After how many hours will the second ship be exactly north of ship A? A. 3.68 C. 5.12 B. 4.03* D. 4.83

48. The sides of a triangular lot are 130 m., 180 m and 190 m. the lot is to be divided by a line bisecting the longest side and drawn from the opposite vertex. Find the length of the line. A. 120 m C. 125 m* B. 130 m D. 128 m

37. An aerolift airplane can fly at an airspeed of 300 mph. If there is a wind blowing towards the cast at 50mph, what should be the plane’s compass heading in order for its course to be 30°? What will be the plane’s ground speed if it flies in this course? A. 19.7, 307.4 mph C. 21.7, 321.8 mph* B. 20.1, 309.4 mph D. 22.3, 319.2 mph

49. The sides of a triangle are 195, 157 and 210, respectively. What is the area of the triangle? A. 73,250 sq. units C. 14,586 sq. units* B. 10,250 sq. units D. 11,260 sq. units

38. A man finds the angle of elevation of the top of a tower to be 30°. He walks 85 m nearer the tower and finds its angle of elevation to be 60°. What is the height of the tower? A. 76.31 m C. 73.16 m B. 73.31 m D. 73.61 m*

50. The sides of a triangle are 8, 15 and 17 units. If each side is doubled, how many square units will the area of the new triangle be? A. 240* C. 320 B. 420 D. 200

39. A pole cast a shadow 15 m long when the angle of elevation of the sun is 61°. If the pole is leaned 15° from the vertical directly towards the sun, determine the length of the pole. A. 54.23 m* C. 42.44 m B. 48.23 m D. 46.21 m

51. Find the supplement of an angle whose compliment is 62°. A. 28° C. 152°* B. 118° D. None of these 52. A certain angle has a supplement 5 times its compliment. Find the angle. A. 67.5° * C. 168.5° B. 157.5° D. 186°

40. When supporting a pole is fastened to it 20 feet from the ground 15 feet from the pole. Determine the length of the wire and the angle it makes with the pole. A. 24 ft, 53.13° C. 24 ft, 53.13° B. 24 ft, 36.87° D. 25 ft, 36.87°*

53. The sum of the two interior angles of the triangle is equal to the third angle and the difference of the two angles is equal to 2/3 of the third angle. Find the third angle. A. 15° C. 90°* B. 75° D. 120°

41. When supporting a pole is fastened to it 20 feet from the ground 15 feet from the pole. Determine the length of the wire and the angle it makes with the pole. A. 24 ft, 53.13° C. 24 ft, 53.13° B. 24 ft, 36.87° D. 25 ft, 36.87°*

54. The measure 0f 1 ½ revolutions counter-clockwise is: A. 540° * C. +90° B. 520° D. -90°

42. Points A and B are 100 m apart and are of the same elevation as the foot of a building. The angles of elevation of the top of the building from points A and B are 21° and 32° respectively. How far is A from the building in meters.? A. 259.28* C. 271.64 B. 265.42 D. 277.29

55. The measure of 2.25 revolutions counterclockwise is: A. -835° C. 805° B. -810° D. 810°* 56. Solve for θ: sin θ – sec θ + csc θ – tan 2θ = –0.0866 A. 40° C. 47° B. 41° D. 43°*

43. The captain of a ship views the top of a lighthouse at an angle of 60° with the horizontal at an elevation of 6 meters above sea level. Five minutes later, the same captain of the ship views the top of the same lighthouse at an angle of 30° with the horizontal. Determine the speed of the ship if the lighthouse is known to be 50 meters above sea level. A. 0.265 m/sec C. 0.169 m/sec* B. 0.155 m/sec D. 0.210 m/sec

57. What are the exact values of the cosine and tangent trigonometric functions of acute angle A, given that sin A = 3/7?

44. An observer wishes to determine the height of a tower. He takes sights at the top of the tower from A and B, which are 50 feet apart, at the same elevation on a direct line with

2

MECHANICAL ENGINEERING COMPREHENSIVE EVALAUTION COURSE 1 TOPIC: PLANE AND SPHERICAL TRIGONOMETRY B. -1/3 D. 2/3* 58. Given three angles A, B, and C whose sum is 180°. If the tan A + tan B + tan C = x, find the value of tan A x tan B x 75. If sin Ѳ – cos Ѳ = -1/3, what is the value of sin 2 Ѳ? tan C. A. 1/3 C. 8/9* A. 1 – x C. x/2 B. 1/9 D. 4/9 B.√x D. x* 76. If x cos Ѳ + y sin Ѳ = 1 and x sin Ѳ – y cos Ѳ = 3, what is 59. What is the sine of 820°? the relationship between x and y? A. 0.984 * C. 0.866 A. x2 + y2 = 20 C. x2 + y2 = 16 B. -0.866 D. -0.500 B. x2 – y2 = 5 D. x2 + y2 = 10* 60. csc 270° = ? A. -√3 B. -1 *

77. If sin x + 1/sin x = √2 , then sin2 x + 1/sin2 x is equal to: A. √2 C. 2 B. 1 D. 0*

C. √3 D. 1

61. If coversine Ѳ is 0.134, find the value of Ѳ. A. 60° * C. 30° B. 45° D. 20°

78. The equation 2 sin Ѳ + 2 cos Ѳ – 1 = √3 is: A. An identity C. A conditional equation* B. A parametric equation D. A quadratic equation

62. 62. Solve for cos 72° if the given relationship is cos 2A = 2 cos2 A – 1. A. 0.309* C. 0.268 B. 0.258 D. 0.315

79. If x + y = 90°, then ((sin x tan y)/(sin y tan x)) is equal to: A. tan x C. cot x* B. cos x D. sin x

63. 63. If sin 3A = cos 6B then: A. A + B = 180° B. A + 2B = 30°*

80. if cos Ѳ = x / 2 then 1 – tan2 Ѳ is equal to: A. (2×2 + 4)/x2 C. (2×2 – 4)/x B. (4 – 2×2)/x2 D. (2×2 – 4)/x2*

C. A – 2B = 30° D. A + B = 30°

64. Find the value of sin (arcos 15/17). A. 8/17* C. 8/21 B. 17/9 D. 8/9

81. Find the value in degrees of arcos (tan 24°). A. 61.48 C. 63.56* B. 62.35 D. 60.84

65. Find the value of cos [arcsin (1/3) + arctan (2/√5)]

82. arctan [2 cos ((√3)/2)] is equal to: A. π/3 C. π/6 B. π/4* D. π/2 83. Solve for x in the equation: arctan (2x) + arctan (x) = π/4 A. 0.821 C. 0.281* B. 0.218 D. 0.182

* 66. If sin 40° + sin 20° = sin Ѳ, find the value of Ѳ. A. 20° C. 120° B. 80°* D. 60°

84. Solve for x from the given trigonometric equation: arctan (1 – x) + arctan (1 + x) = arctan 1/8 A. 4 * C. 8 B. 6 D. 2

67. How many different value of x from 0° to 180° for the equation (2sin x – 1)(cos x + 1) = 0? A. 3 * C. 1 B. 0 D. 2

85. Solve for y if y = (1/sin x – 1/tan x)(1 + cos x) A. sin x* C. tan x B. cos x D. sec2 x

68. For what value of Ѳ (less than 2π) will the following equation be satisfied? sin2 Ѳ + 4sinѲ + 3 = 0 A. π C. 3π/2* B. π/4 D. π/2

86. Solve for x: x = (tan θ + cot θ)2 sin2 θ – tan 2 θ. A. sin θ C. 1* B. cos θ D. 2 87. Solve for x: x = 1 – (sin θ – cos θ)2 A. sin θ cos θ C. cos 2θ B. -2 cos θ D. sin 2θ*

69. Find the value of x in the equation csc x + cot x = 3. A. π/4 C. π/2 B. π/3 D. π/5*

88. Simplify cos4 θ – sin4 θ A. 2 B. 1

70. If sec2 A is 5/2, the quantity 1 – sin2 A is equivalent to: A. 2.5 C. 1.5 B. 0.6 D. 0.4*

C. 2 sin2 θ + 1 D. 2 cos2 θ – 1*

89. Solve for x: x = ((1 – tan2 a)/(1 + tan2 a)) A. cos a C. cos 2a* B. sin 2a D. sin a

71. Find sin x if 2 sin x + 3 cos x – 2 = 0. A. 1 & -5/13* C. 1 & 5/13 B. -1 & 5/13 D. -1 & -5/13

90. which of the following is different from the others? A. 2 cos 2x – 1 C. cos 3x – sin 3x* B. cos 4x – sin 4x D. 1 – 2 sin 2x

72. If sin A = 4/5, A in quadrant II, sin B = 7/25, B in quadrant I, find sin (A + B). A. 3/5* C. 3/4 B. 2/5 D. 4/5

91. Find the value of y: y = (1 + cos 2θ) tan θ. A. cos θ C. sin 2θ* B. sin θ D. cos 2θ

73. If sin A = 2.571x, cos A = 3.06, and sin 2A = 3.939x, find the value of x. A. 0.350 C. 0.100 B. 0.250* D. 0.150

92. The equation 2 sinh x cosh x is equal to: A. ex C. sinh 2x* B.e-x D. cosh 2x

74. If cos Ѳ = √3/2, what is the value of x if x = 1 – tan2 Ѳ. A. -2 C. 4/3

93. Simplifying the equation sin2 θ(1 + cot2 θ)

3

A. 1* B. sin2 θ

MECHANICAL ENGINEERING COMPREHENSIVE EVALAUTION COURSE 1 TOPIC: PLANE AND SPHERICAL TRIGONOMETRY 2 C. sin θ sec2 θ A. 36.8 meters D. sin2 θ B. 37.1 meters

94. If tan θ = x2, which of the following is incorrect? A. sin θ = 1/√(1 + x4) * C. cos θ = 1/√(1 + x4) B. sec θ = √(1 + x4) D. csc θ = √(1 + x4) / x2

C. 36.3 meters* D. 37.4 meters

109. The area of the triangle whose angles are 61°9’32”, 34°14’46”, and 84°35’42” is 680.60. the length of the longest side is: A. 35.53 C. 52.43* B. 54.32 D. 62.54

95. In an isosceles right triangle, the hypotenuse is how much longer than its sides? A. 2 times C. 1.5 times B. √2 times * D. None of these

110. Given a triangle ABC whose angles are A = 40°, B = 95° and side b = 30 cm. find the length of the bisector of angle C. A. 21.74 cm* C. 20.45 cm B. 22.35 cm D. 20.98 cm

96. Find the angle in mils subtended by a line 10 yards long at a distance of 5000 yards. A. 2.5 mils C. 4 mils B. 2 mils* D. 1 mil

111. The sides of a triangular lot are 130 m, 180 m, and 190 m. the lot is to be divided by a line bisecting the longest side and drawn from the opposite vertex. The length of this dividing line is: A. 100 meters C. 125 meters* B. 130 meters D. 115 meters

97. The angle or inclination of ascend of a road having 8.25% grade is _____degrees. A. 5.12 degrees C. 1.86 degrees B. 4.72 degrees* D. 4.27 degrees 98. The sides of a right triangle is in arithmetic progression whose common difference if 6 cm. its area is: A. 216 cm2 * C. 360 cm2 B. 270 cm2 D. 144 cm2

112. From a point outside of an equilateral triangle, the distances to the vertices are 10m, 10m, and 18m. Find the dimension of the triangle. A. 25.63 C. 19.94* B. 45.68 D. 12.25

99. The hypotenuse of a right triangle is 34 cm. Find the length of the shortest leg if it is 14 cm shorter than the other leg. A. 15 cm C. 17 cm B. 16 cm* D. 18 cm

113. Points A and B 1000 m apart are plotted on a straight highway running East and West. From A, the bearing of a tower C is 32 degrees N of W and from B the bearing of C is 26 degrees N of E. Approximate the shortest distance of tower C to the highway. A. 264 meters C. 284 meters B. 274 meters* D. 294 meters

100. A truck travels from point M northward for 30 min. then eastward for one hour, then shifted N 30° W. if the constant speed is 40 Kph, how far directly from M, in km. will be it after 2 hours? A. 43.5 C. 47.9* B. 45.2 D. 41.6

114. An airplane leaves an aircraft carrier and flies South at 350 mph. The carrier travels S 30° E at 25 mph. If the wireless communication range of the airplane is 700 miles, when will it lose contact with the carrier? A. after 4.36 hours C. after 2.13 hours* B. after 5.57 hours D. after 4.54 hours

101. Two sides of a triangle measures 6 cm. and 8 cm. and their included angle is 40°. Find the third side. A. 5.144 cm* C. 4.256 cm B. 5.263 cm D. 5.645 cm

115. A statue 2 meters high stands on a column that is 3 meters high. An observer in level with the top of the statue observed that the column and the statue subtend the same angle. How far is the observer from the statue? A. 5√2 meters C. 20 meters B. 2√5 meters* D. √10 meters

102. Given a triangle: C = 100°, a = 15, b = 20. Find c: A. 34 C. 43 B. 27* D. 35 103. Given angle A = 32°, angle B = 70°, and side c = 27 units. Solve for side a of the triangle. A. 24 units C. 14.63 units* B. 10 units D. 12 units

116. From the top of a building 100 m high, the angle of depression of a point A due East of it is 30°. From a point B due South of the building, the angle of elevation of the top is 60°. Find the distance AB. A. 100 + 3√30 C. 100 (√30) / 3 B. 200 – √30 D. 100√3/ 30*

104. In a triangle, find the side c if the angle C = 100°, side b = 20, and side a = 15. A. 28 C. 29 B. 27* D. 26

117. An observer found the angle of elevation of the top of the tree to be 27°. After moving 10m closer (on the same vertical and horizontal plane as the tree), the angle of elevation becomes 54°. Find the height of the tree. A. 8.65 meters C. 7.02 meters B. 7.53 meters D. 8.09 meters*

105. Two sides of a triangle are 50 m. and 60 m. long. The angle included between these sides is 30 degrees. What is the interior angle (in degrees) opposite the longest side? A. 92.74 C. 94.74 B. 93.74* D. 91.74 106. The sides of a triangle ABC are AB = 15 cm, BC = 18 cm, and CA = 24 cm. Determine the distance from the point of intersection of the angular bisectors to side AB. A. 5.21 cm C. 4.73 cm* B. 3.78 cm D. 6.25 cm

118. From a point A at the foot of the mountain, the angle of elevation of the top B is 60°. After ascending the mountain one (1) mile to an inclination of 30° to the horizon, and reaching a point C, an observer finds that the angle ACB is 135°. A. 14386 C. 11672 B. 12493* D. 11223

107. If AB = 15 m, BC = 18 m and CA = 24 m, find the point of intersection of the angular bisector from the vertex C. A. 11.3 C. 13.4 B. 12.1 D. 14.3*

119. A vertical pole is 10 m from a building. When the angle of elevation of the sum is 45°, the pole cast a shadow on the building 1 m high. Find the height of the pole. A. 0 meter C. 12 meters B. 11 meters * D. 13 meters

108. In triangle ABC, angle C = 70 degrees; angle A = 45 degrees; AB = 40 m. what is the length of the median drawn from vertex A to side BC?

4

MECHANICAL ENGINEERING COMPREHENSIVE EVALAUTION COURSE 1 TOPIC: PLANE AND SPHERICAL TRIGONOMETRY 120. A pole cast a shadow of 15 meters long when the angle of 131. A plane hillside is inclined at an angle of 28° with the elevation of the sun is 61°. If the pole has leaned 15° from horizontal. A man wearing skis can climb this hillside by the vertical directly toward the sun, what is the length of following a straight path inclined at an angle of 12° to the the pole? horizontal, but one without skis must follow a path inclined A. 52.43 meters C. 53.25 meters at an angle of only 5° with the horizontal. Find the angle B. 54.23 meters* D. 53.24 meters between the directions of the two paths. A. 13.21° C. 15.56°* 121. An observer wishes to determine the height of a tower. He B. 18.74° D. 17.22° takes sights at the top of the tower from A and B, which are 50 ft. apart, at the same elevation on a direct line with the 132. If circumference of a circle is divided into 360 congruent tower. The vertical angle at point A is 30° and at point B is parts, angle subtended by one part at center of circle is 40°. What is the height of the tower? called A. 85.6 feet C. 110.29 feet A. Angle C. radian B. 143.97 feet D. 92.54 feet* B. Degree* D. minute 122. From the top of tower A, the angle of elevation of the top of the tower B is 46°. From the foot of tower B the angle of elevation of the top of tower A is 28°. Both towers are on a level ground. If the height of tower B is 120m, how high is tower A in m? A. 38.6 C. 44.1 B. 42.3 D. 40.7*

133. Vertex of an angle in standard form is at A. (1,0) C. (0,1) B. (1,1) D. (0,0)*

123. Points A and B are 100 m apart and are on the same elevation as the foot of a building. The angles of elevation of the top of the building from points A and B are 21° and 32°, respectively. How far is A from the building in m? A. 271.6 C. 259.2* B. 265.4 D. 277.9

135. 1 radian = A. 57°17′45″* B. 180°

134. In one hour, minutes hand of a clock turns through A. 5π/6 radians C. 4π/9 radians B. π/4 radians D. 180π radians* C. 1° D. 180′

136. Central angle of an arc of a circle whose length is equal to radius of circle is called the A. Degree C. radian* B. Minute D. second

124. A man finds the angle of elevation of the top of a tower to be 30 degrees. He walks 85 m. nearer the tower and finds its angle of elevation to be 60 degrees. What is the height of the tower? A. 76.31 meters C. 73.31 meters B. 73.61 meters* D. 73.16 meters

137. System of measurement in which angle is measured in radians called the A. Circular system* C. Sexagesimal system B. MKS system D. CGS system

125. The angle of elevation of a point C from a pint B is 29°42’; the angle of elevation of C from another point A 31.2 m directly below B is 59°23’. How high is C from the horizontal line through A? A. 47.1 meters* C. 35.1 meters B. 52.3 meters D. 66.9 meters

138. System of measurement in which angle is measured in degrees, and its sub-units, minutes and seconds is called the A. Circular system C. Sexagesimal system* B. MKS system D. CGS system 139. 60th part of one degree is called one A. Second C. radian B. Degree D. minute*

126. A rectangular piece of land 40m x 30m is to be crossed diagonally by a 10-m wide roadway. If the land cost P1,500.00 per square meter, the cost of the roadway is: A. P401.10 C. P601,650.00* B. P60,165.00 D. P651,500.00

140. Union of two non-collinear rays which have a common endpoint is called the A. Angle* C. radian B. Degree D. minute

127. A man improvises a temporary shield from the sun using a triangular piece of wood with dimensions of 1.4m, 1.5 m, and 1.3 m. with the longer side lying horizontally on the ground, he props up the other corner of the triangle with a vertical pole 0.9m long. What would be the area of the shadow on the ground when the sun is vertically overhead? A. 0.5 m2 * C. 0.84 m2 B. 0.75 m2 D. 0.95 m2

141. A ladder leans against wall at point B (window end) from a ground level and makes an angle horizontally at 52°. height of ladder is 15 m. When same ladder leans above point B at point A (window start) and makes an angle of 85° horizontally. distance between point A and point B is A. 12.4 m C. 5 m B. 4.12 m D. 3.12 m*

128. A rectangular piece of wood 4 cm x 12 cm tall is titled at an angle of 45°. Find the vertical distance between the lower corner and the upper corner. A. 4√2 cm C. 8√2 cm* B. 2√2 cm D. 6√2 cm

142. Height of a light house is 65 m. angles of elevation and depression of top and foot of a radar mast are 52° and 30° respectively. height of radar mast is A. 102 m C. 305 m B. 209.09 m* D. 109.09 m

129. A clock has a dial face 12 inches in radius. The minute hand is 9 inches long while the hour hand is 6 inches long. The plane of rotation of the minute hand is 2 inches above the plane of rotation of the hour hand. Find the distance between the tips of the hands at 5:40 AM. A. 9.17 inches* C. 10.65 inches B. 8.23 inches D. 11.25 inches

143. A house is built at top of cliff. From foot of cliff at a distance of 75 m, angle of elevation of top of house is 51° and angle of elevation at top of cliff 41°. height of house is A. 38.25 m C. 36 m B. 32.42 m D. 27.42 m* 144. From top of mountain 68 m high, angle of depression of two boats due east of it are 52° and 26° respectively. distance between ships is A. 58.28 m C. 86.29 m* B. 68.29 m D. 53.72 m

130. If the bearing of A from B is 40° W, then the bearing of B from A is: A. N 40° E * C. N 50° E B. N 40° W D. N 50° W

5

MECHANICAL ENGINEERING COMPREHENSIVE EVALAUTION COURSE 1 TOPIC: PLANE AND SPHERICAL TRIGONOMETRY 145. Express 45° in mils. building B is 120 m high, determine the height of building A. 50 C. 200 A. assume both buildings lie in the same horizontal plane. B. 800* D. 112 A. 75.91 m C. 57.30 m B. 73.21 m D. 79.29 m* 146. What is the value in degrees for 1 rad? A. 57.3°* C. 0.159° 162. The angle of elevation of a tower at a place A south of it is B. 62.8° D. 114.59° 30°, and B west of A and a distance of 50 m from the angle of elevation is 18°, determine the height of the tower. 147. How many radians is equivalent to 100 grads? A. 19.65 m* C. 22.22 m A. /2* C. /3 B. 15.42 m D. 26.32 m B. 3 /4 D. 2/3 163. A tower and a monument stand on a level plane. The angle of depression of the top and bottom of the monument 148. Find the supplement of an angle whose complement is 62°. viewed from the top of the tower are 13° and 31°, A. 152°* C. 28° respectively. The height of the tower is 145 ft. Find the B. 75° D. 87° height of the monument. A. 75.7 ft C. 92.2 ft 149. Find the angle equal to 3/7 of its supplement. B. 89.3 ft* D. 98.6 ft A. 54° C. 36° B. 24° D. 27°* 164. A pole stands on a plane which makes an angle of 15° with the horizontal. A wire from the top of the pole is anchored 150. If the supplement of a given angle is 5/2 of its complement, on a point 8 meters from the foot of the pole. If the angle find the value of the angle. between the wire and the plane is 30°, find the length of A. 50° C. 30°* the wire. B. 20° D. 40° A. 10.93 m * C. 12.56 m B. 11.62 m D. 13.29 m 151. A certain angle has an explement 5 times the supplement, find the angle. 165. The sides of a triangle are in a ratio 4:5:6. The smallest A. 120° C. 165° angle is ____. B. 135°* D. 150° A. 82° C. 69° B. 56° D. 41°* 152. Simplify sec θ – sec θ sin2 θ A. sin θ C. cos θ* 166. The sides of a triangle are 40, 50 and 70 cm, respectively. B. cos2 θ D. sin2 θ Find the length of the bisector of the largest angle. sin4 𝑥−cos4 𝑥 A. 28.13* C. 20.52 153. Simplify 2 sin 𝑥−cos2 𝑥 B. 26.32 D. 28.40 A. 1 * C. sin x B. sec x D. 3/2 167. A pole which leans to the sun by 10°15’ from the vertical casts a shadow of 9.43m on the level ground when the sin 𝑥 tan 𝑦 154. If x + y = 90°, then is equal to _____. angle of elevation of the sun is 54°50’. Find the length of sin 𝑦 tan 𝑥 the pole. A. tan y* C. cot y A. 15.6 m C. 17.7 m B. –tan y D. –cot x B. 18.3 m* D. 116.9 m 155. The expression A. B.

1−sin 𝑥 cos 𝑥 1+cos 𝑥 sin 𝑥

1−sin 𝑥 cos 𝑥

is equal to _____. C. D.

168. From the top of a hill the angles of depression of the top and bottom of a flagpole 25 ft high at the foolt of the hill are observed to be 45°13’ and 47°12’, respectively. Find the height of the hill. A. 334.4 ft C. 320.2 ft B. 410.5 ft D. 373.2 ft*

sin 𝑥 1−sin 𝑥 cos 𝑥 1+sin 𝑥

*

156. Is sec A = -5/4 and A is in the second quadrant, find tan 2A. A. -24/7* C. 7/24 B. 12/25 D. -25/12

169. From a boat sailing due north at 16.5 kph, a wrecked ship and an observation tower are observed in a line due east. One hour later the wrecked ship and the tower have bearings S 34°40’ E and S 65°10’ E. Find the distance between the wrecked ship and the tower. A. 24.25 km* C. 42.71 km B. 27.62 km D. 11.18 km

157. If sin A = 3/5 and A is in the second quadrant while cos B = 7/25 and B is in the first quadrant, find sin (A + B). A. 4/5 C. -3/5* B. -4/5 D. 2/5 158. If sin A = 3/5 and A is in the first quadrant while cos B = 7/25 and B is on the third quadrant, find cos (A + B). A. -117/125* C. -44/125 B. 3/5 D. 117/125

170. Two points X and Y, 1000m apart are located on a straight road running east and west. From X, the bearing of a tower is 32° W of N and from Y the bearing of the tower is 26° N of E. determine the shortest distance of the tower from the road. A. 331 m C. 374 m* B. 415 m D. 443 m

159. The value of tan (A + B), where tan A = 1/3 and tan B = ¼ is _____. A and B are acute angles. A. 1/11 C. 7/13 B. 11/12 D. 7/11*

171. A plane travels 500 miles due east from A to B, then banks to the right and travels 1000 miles to C and finally travels 1200 miles from C back to A. Find the bearing of the course taken by the airplane from B to C. A. N 11.95° E C. S 10.95° E* B. S 12.05° E D. N 10.56° E

160. A man found that the angle of elevation of the top of the tower to be 30°. He walks 85 m nearer the tower and finds its angle of elevation to be 60°. What is the height of the tower? A. 65.43 m C. 87.76 m B. 73.61 m* D. 80.13 m

172. Find the area of the excircle, which is tangent to side AB of triangle ABC where AB = 12 cm, BC = 24 cm, and CA = 32 cm. A. 63.4 C. 97.1

161. The angle of elevation of the top of a building B from the top of building A is 28° and the angle of elevation of the top of building A from the base of building B is 46°. If

6

B. 72.1 *

MECHANICAL ENGINEERING COMPREHENSIVE EVALAUTION COURSE 1 TOPIC: PLANE AND SPHERICAL TRIGONOMETRY D. 83.2

173. A circle is inscribed in an isosceles triangle whose base is 16 cm and whose altitude is 15 cm. Determine the area of the said circle. A. 66.10 cm2 C. 62.32 cm2 B. 72.38 cm2 * D. 73.32 cm2

A. 25 cm2 * B. 5 cm2

174. Three circles with radii 3, 5 and 9 cm, respectively, are externally tangent. What is the area of the triangle formed by joining their centers? A. 48 cm2* C. 52 cm2 B. 44 cm2 D. 50 cm2

C. 10 cm2 D. 100 cm2

186. Find the area of the shaded portion as shown :

6 cm

175. A triangular piece of wood has an area of 375 cm2. Two of its angles are 16° and 56°. Find the length of the longest side. A. 62.52 cm C. 48.32 cm B. 52.61 cm D. 55.87 cm*

r

8 cm

6 cm

176. The sides of a triangle are 14 cm, 15 cm, and 13 cm, respectively. What is the area of the circumscribing circle? A. 202.2 C. 184.3 B. 110.5 D. 207.4*

R

A. 65  B. 81 

B. 56 * D. 25 

187. Find the radius of the small circle as shown :

177. The sides of a right triangle are in arithmetic progression whose common difference is 6. Find the hypotenuse. A. 36 C. 18 B. 30* D. 24

4m 8m

178. A ladder 5 m long leans on a wall and makes an angle of 30° with the horizontal. Find the vertical height from the top to the ground. A. 2.50 m* C. 1.50 m B. 2.00 m D. 2.75 m

r A. 0.385 m B. 0.581 m

C. 0.686 m* D. 0.985 m

188. If tan x = 1/2 , tan y = 1/3 , What is the value of tan ( x +y)? A. 1* C. 3 B. 2 D. 4

179. In a triangle, find the side c if angle C = 100° , side b = 20 and side a = 15. A. 28 C. 29 B. 27* D. 26

189. Simplify cos(30 degrees – A) – cos(30 degrees + A) as a function of angle A only. A. tan A C. cos A B. sec A D. sin A*

180. A PLDT tower and a monument stand on a level plane . The angles of depression of the top and bottom of the monument viewed from the top of the PLDT tower are 13° and 35° respectively. The height of the tower is 50 m. Find the height of the monument. A. 33.51 m* C. 47.30 m B. 7.58 m D. 30.57 m

190. Find the area of the excircle, which is tangent to side AB of triangle ABC where AB = 12 cm, BC = 24 cm, and CA = 32 cm. A. 63.4 C. 97.1 B. 72.1 * D. 83.2

181. A man finds the angle of elevation of the top of a tower to be 30 degrees. He walks 85 m nearer the tower and find its angle of elevation to be 60 degrees. What is the height of the tower ? A. 76.31 m C. 73.16 m B. 73.31 m D. 73.61 m*

191. A circle is inscribed in an isosceles triangle whose base is 16 cm and whose altitude is 15 cm. Determine the area of the said circle. A. 66.10 cm2 C. 62.32 cm2 B. 72.38 cm2 * D. 73.32 cm2

182. The sides of a triangle are 195, 157, and 210 respectively. What is the area of the triangle ? A. 73 250 sq. units C. 14 586 sq. units* B. 10 250 sq. units D. 11 260 sq. units

192. Three circles with radii 3, 5 and 9 cm, respectively, are externally tangent. What is the area of the triangle formed by joining their centers? A. 48 cm2* C. 52 cm2 B. 44 cm2 D. 50 cm2

183. In triangle BCD, BC = 25 m, and CD = 10 m. The perimeter of the triangle maybe : A. 69 m* C. 71 m B. 70 m D. 72 m

193. A triangular piece of wood has an area of 375 cm2. Two of its angles are 16° and 56°. Find the length of the longest side. A. 62.52 cm C. 48.32 cm B. 52.61 cm D. 55.87 cm*

184. The sides of a triangle are 8 cm , 10 cm, and 14 cm. Determine the radius of the inscribed and circumscribing circle. A. 3.45, 7.14 C. 2.45, 8.14 B. 2.45, 7.14* D. 3.45, 8.14

194. The sides of a triangle are 14 cm, 15 cm, and 13 cm, respectively. What is the area of the circumscribing circle? A. 202.2 C. 184.3 B. 110.5 D. 207.4*

185. Find the area of the shaded portion of the two concentric circles whose chord outside the small circle is 10 cm.

195. The spherical excess of a spherical triangle with A = 56°, B = 90°, and C = 102° is ___. A. 68° * C. 96° B. 78° D. 86°

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MECHANICAL ENGINEERING COMPREHENSIVE EVALAUTION COURSE 1 TOPIC: PLANE AND SPHERICAL TRIGONOMETRY 211. Solve for a in a spherical triangle given b=30°15’, c=68°27’, 196. A spherical triangle has the following angles: A = 79°, B = A=126°43’. 66°, and C = 115°. Solve for the spherical excess. A. 27°57’ C. 67°31’ A. 70° C. 80°* B. 87°52’* D. 57°12’ B. 75° D. 85° 212. Solve for b in a spherical triangle given A = 55°17’, B = 197. What is the area of a spherical triangle whose angles are 77°28’, and C = 97°43’. 123°, 84°, and 73°? The radius of the sphere is 30m. A. 82.15° C. 55.21° A. 1480 m2 C. 1959 m2 B. 70.51° D. 80.05°* B. 1571 m2 * D. 1863 m2 213. Solve for A in a spherical triangle given a = 86°20’, b = 198. The radius of the spherical triangle on a sphere is 15 cm 45°30’, and C = 120°45’. and the angles are A = 95°, B = 73°, and C = 130°. Solve A. 66°11’ C. 20°37’ for the area of the spherical triangle. B. 24°34’ D. 64°49’* A. 632.384 cm2 C. 433.458 cm2 B. 628.564 cm2 D. 463.385 cm2* 214. Solve for C in a spherical triangle given a = 59°38’, c = 130°38’, and B = 98°40’. 199. Find the area of a spherical triangle on a sphere of radius A. 133°39’11’’ C. 119°21’51’’ 10 inches, if A = 132°, B = 138°, and C = 150°. B. 95°47’31’’ D. 123°53’23’’* A. 439.549 in2 C. 418.879 in2* B. 485.225 in2 D. 460.125 in2 215. Solve for B in the spherical triangle having a = 57°24’, b = 69°17’, and A = 45°38’. 200. Given the right spherical triangle with A = 70°45’; B = A. 52°32’ * C. 35°24’ 149°10, solve for a. B. 25°23’ D. 17°26’ A. 49°58’ * C. 63°22’ B. 29°52’ D. 36°16’ 216. In the following oblique spherical triangle given that: B=65°33’, b=64°23’15”, a=99°40’4”, solve for A. 201. Given the right spherical triangle with A = 63°15’ and B = A. 44°11’25” C. 84°21’10” * 135°43’, solve for side b. B. 64°23’12” D. 104°43’23” A. 143°17’13’’* C. 122°39’53’’ B. 132°19’33’’ D. 151°15’23’’ 217. Solve for B in a spherical triangle given A = 118°52’38’’, C = 28°24’14’’, and b = 100°50’. 202. Given the right spherical triangle with a = 24°52’ and A = A. 61°44’34.22’’ C. 72°38’42.11’’ 38°24’, solve for b. B. 69°43’34.08’’* D. 75°21’25.28’’ A. 114°27’ C. 144°13’* B. 151°56’ D. 122°33’ 218. Solve for b in a spherical triangle given A = 43°27’, B = 69°33’, and C = 105°11’. 203. Given the right spherical triangle with A = 60.5° and B = A. 76.12° * C. 83.25° 135.5°, solve for c. B. 79.41° D. 89.09° A. 115.24° C. 175.32° B. 125.15° * D. 143.12° 219. In a spherical triangle given a = 64°22’, b = 73°01’, and C = 102°33’, solve for A. 204. Given a spherical triangle whose parts are a = 73°, b = 62°, A. 45°21’ C. 61°50’ * and C = 90°. Solve for A. B. 58°37’ D. 75°33’ A. 74.89° * C. 69.24° B. 71.52° D. 47.92° 220. In spherical trigonometry, 1 nautical mile is equivalent to ____ feet. 205. Given the right spherical triangle ABC given that a = 87°16’; A. 1609 C. 5280 B = 38°45’; and C = 0°, solve for c. B. 2567 D. 6080* A. 82°42’9’’ C. 87°52’3’’* B. 77°22’4’’ D. 91°32’7’’ 221. In spherical trigonometry, 1 statute mile is equivalent to ____ feet. 206. Given the right spherical triangle with A = 48°32’ and B = A. 1609 C. 5280* 123°47’, solve for side b. B. 2567 D. 6080 A. 135.61° C. 127.31° B. 137.91° * D. 115.81° 222. The section of the surface of a sphere made by any plane is a ___. 207. For an oblique spherical triangle, find side ‘a’ given that b A. Circle* C. hyperbola = 7.5 m, c = 5.9 m and A = 49°. B. Ellipse D. parabola A. 2.33 m C. 5.73 m* B. 9.26 m D. 4.29 m 223. This term refers to the angle between the north vector and the perpendicular projection of the start down onto the For numbers 203 – 205: horizon. Solve the following oblique spherical triangle given that: b = A. Azimuth * C. Solstice 156°12’, c = 112°48’, A = 76°32’. B. Declination D. Zenith 208. Solve for a. A. 62°42’12” B. 55°24’24”

C. 63°49’34” * D. 47°21’51”

209. Solve for B. A. 113°1’2.33” B. 154°4’2.63” *

C. 161°7’1.53” D. 135°2’7.23”

210. Solve for C. A. 87°26’32.52” * B. 77°22’31.44”

C. 65°32’11.74” D. 42°25’32.44”

224. The term refers to an imaginary point directly above a particular location, on the imaginary celestial sphere. A. Azimuth C. Solstice B. Declination D. Zenith* 225. The two points where the rotation axis meets the surface of the Earth are known as the North Pole and the South Pole and the great circle perpendicular to the rotation axis and lying halfway between the poles is known as the ___. A. Meridian C. Parallel B. Equator * D. Bisector

8

MECHANICAL ENGINEERING COMPREHENSIVE EVALAUTION COURSE 1 TOPIC: PLANE AND SPHERICAL TRIGONOMETRY 226. It refers to the angular distance between the local meridian 239. Which of the following is not true about a sphere. (which passes through the point) and the Greenwich A. The section of the surface of a sphere by a plane is meridian (which passes through the Royal Greenwich called a great circle if the plane passes through the Observation in London). center of the sphere. A. Longitude* C. Co-latitude B. The radius of a great circle is equal to the radius of the B. Latitude D. Pole sphere. C. The diameter of a small circle is equal to the radius of 227. It refers to the angular distance north or south of the a great circle. * equator, measured along the meridian through the point. D. The section of the surface of a sphere by a plane is A. Longitude C. Co-latitude called a small circle if the plane does not pass through B. Latitude * D. Pole the center of the sphere. 228. It refers to the angular distance between a point and the closest pole as measured along the meridian passing through the point. A. Longitude C. Co-latitude* B. Latitude D. Pole

240. Which of the following is/are true about spherical triangles? I. The angles at the base of an isosceles spherical triangle are equal. II. If two angles of a spherical triangle are equal, the opposite sides are also equal. III. The angles at the base of a right spherical triangle are equal.

229. In spherical trigonometry, meridians are defined as ____. A. Great circles which pass through the North Pole and South Pole. * B. Small circles which lie parallel to the equator. C. Great circles which lie parallel to the equator. D. Small circle which pass through the North Pole and South Pole.

A. B.

C. I and II only* D. II and III only

241. Which of the following is/are true about spherical triangle? I. If one triangle be the polar triangle of another, the latter will be the polar triangle of the former. II. The three angles of a spherical triangle are together greater than two right angles and less than six right angles. III. If one angle of a spherical triangle be greater than another, the side opposite the greater angle is greater than the side opposite the lesser angle.

230. In spherical trigonometry, parallels are defined as ____. A. Great circles which pass through the North Pole and South Pole. B. Small circles which lie parallel to the equator. * C. Great circles which lie parallel to the equator. D. Small circle which pass through the North Pole and South Pole.

A. B.

231. In spherical trigonometry, the angle of inclination of the planes of the circles is ___. A. The value of the excess of the spherical triangle. B. The half of the value of one small circle. C. The angle between two small circles D. The angle between two great circles. *

I and II only I and III only

C. II and III only D, I, II, and III*

242. Which of the following is/are true about the poles of a circle? I. The poles of a great circle are equally distant from the plane of the circle. II. The poles of a small circle are not equally distant from the plane of the circle. III. The poles of both great circle and small circle are not equally distant from the plane of the circle.

232. If one side of a spherical triangle bet greater than another, the angle opposite the greater side is ___ than/to the angle opposite the less side. A. Lesser C. equal B. Greater * D. always half 233. The sum of the three sides of than/to the circumference of a A. Twice B. Equal

I, II, and III I and III only

A. B.

a spherical triangle is ____ great circle. C. greater D. lesser*

I and II only* I and III only

C. II and III only D. I, II, and III

243. The following are true about spherical triangles except: A. The three arcs pf great circles which form a spherical triangle are called the sides of the spherical triangle. B. The three arcs of a great circle which form a spherical triangle are called the angles of the spherical triangle. * C. The angles formed by the arcs at the point where they meet are called the angles of the spherical triangle. D. The sides and angles of the polar triangle are respectively the supplements of the angles and sides of the primitive triangle.

234. The angle subtended at the center of a sphere by the arc of a great circle which joins the poles of two great circles is ___ to the inclination of the planes of the great circles. A. Twice C. greater B. Equal * D. lesser 235. The sum of the angles of a spherical triangle is ___ than/to 180°. A. Twice C. greater* B. Equal D. lesser

244. The following are true about spherical triangles except: A. Two great circles bisect each other. B. Great circles which passes through the poles of a great circle are called secondaries to that circle. C. A pole of a circle is equally distant from every point of the circumference of the circle. D. The arc of a great circle which is drawn from a pole of a great circle to any point in its circumference is an octant. *

236. The angle between two circles drawn on a sphere is ____ than/to the angle between their tangents. A. Lesser C. greater B. Equal * D. half 237. The sines of the angles of a spherical triangle are ____ to the sines of the opposite sides. A. Lesser C. inversely proportional B. Greater D. proportional*

245. From a second floor window of a building, the angle of depression of an object on the ground is 35°58’, while from a fifth floor window, 9.75m above the first point of observation the angle of depression is 58°35’. How far is the object from the building?. A. 10.70 m* C. 7.76 m B. 17.10 m D. 6.77 m

238. The following are applications of spherical trigonometry except: A. Navigation C. Stellar map making B. Excavation* D. positions of sunrise and sunset

9

MECHANICAL ENGINEERING COMPREHENSIVE EVALAUTION COURSE 1 TOPIC: PLANE AND SPHERICAL TRIGONOMETRY 246. Two streets intersect at an angle of 63°. A triangular lot has A. Circle frontages of 36.65 m and 51.18 m on the two streets. Find B. Sphere* the length of the third side. A. 45.17 m C. 74.53 m B. 74.15 m D. 47.53 m * 247. A and B are points on the opposite sides of a certain body of water. Another point C is located such that AC = 197 m and BC = 157 m, and angle BAC is 51°. Determine the distance AB. A. 168.56 m C. 186.56 m B. 158.76 m * D. 185.67 m 248. For a given spherical triangle whose given parts are: A = 125°32’, C = 90°, and a = 140°. Find side b, A. 49°21’ C. 36°49’ * B. 52°10’ D. 127°50’ 249. A vertical pole is 10 m from a building. When the angle of elevation of the sun is 45°, the pole cast a shadow on the building 1 m high. Find the height of the pole. A. 0 C. 12 m B. 11 m* D. 13 m 250. Calculate the area of a spherical triangle whose radius is 5 m and whose angles are 40°, 65°, and 110°.. A. 12.34 m2 C. 16.45 m2 2 B. 14.89 m D. 15.26 m2 * 251. A right spherical triangle has an angle C = 90°, a = 50°, and c = 80°. Find the side b. A. 45.33° C. 74.33° * B. 78.66° D. 75.89° 252. If the angles of a triangle are in the ratio 2:4:6; what is the largest angle? A. 30° C. 90° * B. 60° D. 15° 253. Find the largest angle of a triangle, if the sum and difference of two angles are 100° and 20°, respectively. A. 60° C. 40° B. 80° * D. 90° 254. What is the measure of one of the remote angles of a triangle if the exterior angle is 130° and the other remote angle measures 60°? A. 70° * C. 50° B. 30° D. 40° 255. To find the angles of a triangle, given only the lengths of the sides, one would use: A. Law of cosines* C. Law of sines B. Law of tangents D. Orthogonal functions 256. Angle between 90° and 180° has: A. Positive sin and cos B. Negative cot and csc C. Negative sec and tan* D. all functions are negative 257. To find the angles of the triangle, given only the lengths of the sides, one would use: A. Law of cosines* C. Law of tangents B. Law of sines D. The inverse-square law 258. The altitude of the sides of a triangle intersect at the point known as: A. Incenter C. orthocenter* B. Circumcenter D. centroid 259. The center of a circle inscribed inside the triangle and it is the intersection of the three angle bisectors of the triangle. A. Incircle C. Incenter* B. Increment D, Median 260. Body or space bounded by surface every point of which is equidistant from a point within.

10

C. spheroid D. ellipsoid