Trigonometry Questions

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1. ECE Board April 1995 A pole cast a shadow of 15 m. How long when the angle of elevation of the sun is 61 degree if the pole has leaned 15 degree from the vertical directly toward the sun? A. 48.24 B. 23.1 C. 54.23 D. 34.3

B. 21.4 degree C. 13.9 degree D. 18.9 degree 6. ECE Board November 2002 A certain angle has an explement 5 times the supplement, Find the angle. A. 67.5 degree B. 108 degree C. 135 degree D. 58.5 degree

2. ECE Board March 1996 Solve for x in the equation: arctan  (2x) + arctan (x) = . 4 A. 0.2841 B. 0.185. C. 0.218 D. 0.821

7. ECE Board November 2002 Find the height of the tree if the angle of elevation of its top changes from 20 degrees to 40 degrees as the observer advances 23 meters toward the base. A. 13.78 m B. 16.78 m C. 14.78 m D. 15.78m

3. ECE Board March 1996 The hypotenuse of a right triangle is 34 cm. Find the lengths of the two legs if one leg is 14 cm. longer than the other. A. 16 cm, 30 cm B. 13 cm, 27 cm C. 15 cm, 29 cm D. 10 cm, 14 cm

8. ECE Board November 2002 A wheel, 3 ft. in diameter, rolls down an inclined plane 30 degrees with the horizontal. How high is the center of the wheel when it is 5 ft from the base of the plane? A. 4 ft B. 2.5 ft C. 3 ft D. 5 ft

4. ECE Board March 1996 4 If sin A = , A in quadrant II, sin B = 5 7 , B in quadrant I, find sin (A+B) 25 7 A. 5 3 B. 5 2 C. 5 3 D. 4

9. ECE Board November 2002 If the complement of the angle theta is 2/5 of its supplement, then theta is _______. A. 45 degree B. 75 degree C. 60 degree D. 30 degree

5. ECE Board March 1996 If 77 degree + 0.40 x = arctan (cot 0.25x), solve for x.

A. 20 degree

10. ECE Board November 2002

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One side of the right triangle is 15 cm long and the hypotenuse is 10 cm longer than the other side. What is the length of the hypotenuse? A. 13.5 cm B. 6.5 cm C. 12.5 cm D. 16.25 cm

C. 1/5 D. ¼ 16. ECE board April 1999 Sin (B – A) is equal to ____, when B=270 degrees and A is an acute angle. A. –cos A B. cos A C. –sin A D. sin A

11. ECE BOard November 2003 What is the base B of the logarithmic function log 4 = 2/3? A. 8 B. 2 C. 3 D. 4

17. ECE Board April 1999 if sec2A is 5/2, the quantity 1sin2A is equivalent to A. 0.4 B. 0.8 C. 0.6 D. 1.5

12 ECE Boars November 2003 A transmitter with a height of 15 m is located on top of a mountain, which is 3.0 Km high. What is the farthest distance on the surface of the earth that can be seen from the top of the mountain? Take the radius of the earth to be 6400 Km. A. 205 Km. B. 225 km. C. 152 km. D. 200.82 km.

18. ECE Board April 1999

 cosA  4 -  sinA  4 is equal to ____. A. Cos4A B. Sin 2A C. Cos 2A D. Sin 4A 19. ECE Board April 1999 Of what quadrant is A, if sec A is positive and csc A is negative? A. IV B. I C. III D. II

13. ECE Board November 2003 If y = arcsec (negative square root of 2), what is the value of y in degree? A. 75 B. 60 C. 45 D. 135

20. ECE Board April 1999 Angles are measured from the positive horizontal axis, and the positive direction is counterclockwise. What are the values of sin B and cos B in the 4th quadrant? A. sin B > 0 and cos B < 0 B. sin B < 0 and cos B < 0 C. sin B < 0 and cos B > 0 D. sin B > 0 and cos B > 0

14. ECE Board November 2003 The tangent of the angle of a right triangle is 0.75. What is the csc of the angle? A. 1.732 B. 1.333 C. 1.667 D. 1.414 15. ECE Board November 2003 If arctan 2x + arctan 3x = 45 degrees, what is the value of x? A. 1/6 B. 1/3

21. ECE Board November 1997

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Find the value of x in the equation csc x + cot x = 3 π A. 4 π B. 16 π C. 3 π D. 5

If log of 2 to the base 2 plus log of x to the base 2 is equal to 2, then the value of x is A. 2 B. -2 C. -1 D. 3 26. ECE Board November 1998 Solve the equation cos2A = 1- cos2A A. 450 ; 3150 B. 450 ; 2150 C. 450 ; 3450 D. 450 ; 2450

22. ECE Board April 1998 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. 73.61 B. 28 C. 30 .23 D. 82.36

27. ECE Board November 1998 Csc 520 degrees is equal to A. csc 20 degrees B. cos 20 degrees C. tan 45 degrees D. sin 20 degrees 28. ECE Board April 1999 What is 4800 mils equivalent in degrees? A. 2500 B. 2300 C. 2700 D. 2200

23. ECE Board April 1998 Find the angle in mils subtended by a line 10 yards long at a distance of 5,000 yards. A. 2.04 mils B. 10.63 mils C. 10.73 mils D. 4 mils

29. ECE Board April 1999 / November 2000 Cos4 A – sin4 A is equal to _______. A. cos2A B. cos 4A C. sin 2A D. sin 2A

24. ECE Board April 1998 Points A and B, 1000m apart are plotted on straight highway running East and West. From A, the bearing of a tower C is 32 degree West of North and from B, bearing of C is 26 degree North of east. Approximate the shortest distance of tower C to the highway. A. 374 m. B. 364 m. C. 636 m. D. 384 m.

30. ECE Board April 1999 Sin (B – A) is equal to _____. When B = 2700 and A is an acute angle. A. – cos A B. cos A C. –sin A D. sin A

25. ECE Board November 1998

31. ECE Board April 1999

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If sec2A is 5/2, the quantity 1 – sin 2 A is equivalent to A. 0.40 B. 1.5 C. 1.5 D. 1.25

36. ECE Board November 1999 / November 2001 A central angle of 45 subtends an arc of 12 cm. what is the radius of the circles? A. 12.38 cm. B. 15.28 cm. C. 14.28 cm. D. 11.28 cm.

32. ECE Board November 1999 A railroad is to be laid – off in a circular path. What should be the radius if the track is to be change direction by 30 degrees at a distance of 157.08m? A. 300 B. 280 C. 290 D. 350

37. ECE Board November 1999 Given: y = 4cos2X. Determine its amplitude. A. 2 B. 8 C. 2 D. 4

33. ECE Board November 1999 If (2log4 x) – (log 49) = 2, find x. A. 12 B. 15 C. 13 D. 14

38. ECE Board April 2000 If A +B+C = 180 and tan A + tan B + tan C = 5.67, find the value of tan A tan B tan C. A. 5.67 B. 6.15 C. 8.13 D. 9.12

34. ECE Board November 1999 / November 2001 If arctan (X) + arctan (1/3) = /4, the value of x is 1 A. 2 1 B. 4 1 C. 3 1 D. 5

39. ECE Board April 2000 Three times the sine of a certain angle is twice of the square of the cosine of the same angle. Find the angle. A. 30 B. 60 C. 45 D. 10 40. ECE Board April 2001 Solve angle A of an oblique triangle ABC, if a = 25, b = 16 and C = 94.1 degrees. A. 52 degrees and 40 minutes B. 54 degrees and 30 minutes C. 50 degrees and 40 minutes D. 54 degrees and 20 minutes

35. ECE Board November 1999 If tan4A = cot 6A, then what is the value of angle A? A. 9 B. 12 C. 10 D. 14

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A. 3 B. 3 C. 2 D. 2

41. ECE Board April 2003 If tan A = 1/3 and cot B = 2, tan (A-B) is equal to ______________. A. 11/7 B. -1/7 C. -11 / 7 D. 1/7

47. ECE Board April 2004 Given: log (2x -3) = ½. Solve for x if the base is 9. A. 3 B. 12 C. 4 D. 5

42. ECE Board April 2003 Three circles with radii 3, 4 and 5 inches, respectively are tangent to each other externally. Find the largest angle of a triangle formed by joining the centers. A. 72.6 degrees B. 75.1 degrees C. 73.93 degrees D. 73.3 degrees

48. ECE Board November 2004 What is the value of x if log (base x) 1296 = 4? A. 5 B. 3 C. 6 D. 4

43. ECE Board April 2003  sec A tan A  Find the value of if  sec A  tan A 

49. ECE Board April 2001 If sin A = 2.5x and cos A = 5.5x, find the value of A in degrees. A. 54.34 B. 24.44 C. 35.74 D. 45.23

csc A = 2. A. 4 B. 2 C. 3 D. 1

50. ECE Board April 2001 Triangle ABC is a right triangle with right angle at C. If BC = 4 and the altitude to the hypotenuse is a 1, find the area of the triangle ABC. A. 2.43 B. 2.07 C. 2.11 D. 2.70

44. ECE Board November 2003 If Log 2 =x, log 3 = y, what is log 2.4 in terms of x and y? A. 3x + 2y -1 B. 3x + y - 1 C. 3x + y +1 D. 3x – y + 1 45. ECE Board November 2003 Simplify the expression 4 cos y sin y (1 – 2 sin 2 y) A. sec 2y B. cos 2y C. tan 4y D. sin 4y

51. ECE Board April 2001 The measure of 2.25 revolutions counterclockwise is A. 810 B. -810 C. 805 D. -825

46. ECE Board November 2003 If 2 log 3 (base x) + log 2 (base x) =2 + log 6 (base x), then x equals ______.

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52. ECE Board November 2001 If cot 2A cot 68 = 1, then tan A is equal to _____. A. 0.419 B. 491 C. 0.194 D. 194

57. ECE Board November 1991 The captain of a ship views the top of a lighthouse at an angle of 60o 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 300 with the horizontal. Assume that the ship is moving directly away from the lighthouse; determine the speed of the ship. The lighthouse is known to be 50 meters above sea level. Solve the problems by trigonometry. A. 40.16 B. 22.16 m/ min C. 10.16 m/ min D. 12.16 m/ min

53. ECE Board April 2002 / April 1999 Assuming that the earth is a sphere whose radius is 6400 km, find the distance along a 3-degree arc at the equator of the earth’s surface. A. 333.10 km B. 335.10 km C. 533.10 km D. 353.01 km 54. ECE Board November 1998 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 triangle differ by 21 square units A. 4 and 10 B. 4 and 26 C. 6 and 14 D. 7 and 23

58. ECE Board April 1992 Given: P = A sin t + B cos t Q = A cos t – B sin t From the given equation, derive another equation showing the relationship between P, Q and A and B not involving any of the trigonometric function of angle t. A. P 2 - Q2 =A 2 +B2 B. P 2 +Q2 =A 2 +B2 C. P 2 - Q2 =A 2 - B2

55. ECE Board November 1996 If sin A = 2.511x, cos A = 3.06x and sin 2A = 3.969x, find the value of x? A. 0.265 B. 0.256 C. 0.625 D. 0.214

D. P 2 +Q2 =A 2 - B2 59. ECE Board April 1993 Express 45o in mils A. 8000 mils B. 800 mils C. 80 mils D. 8 mils

56. ECE Board April 1991 Find the value of P if it is equal to sin21o +sin2 2o +sin23o +... +sin290o A. 45.5 B. 0 C. infinity D. indeterminate

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60. ECE Board April 1995 A pole cast a shadow of 15 meters long when the angle of elevation of the sun is 61. If the pole has leaned 15 from the vertical directly toward the sun. What is the length of the pole? A. 53.24m B. 54.23m C. 53.32m D. 52.43m

65. ECE Board November 1997 The denominator of a certain fraction is three more than twice the numerator. If 7 is added to both terms of the fraction, the resulting fraction is 3 / 5. Find the original fraction A. 5 / 13 B. 3 / 5 C. 4 / 5 D. 3 / 8

61. ECE Board April 2000 If A + B + C =180 and tan A + tan B +Tan C = 5.67, find the value of tan A tan B tan C A. 5.67 B. 1.78 C. 6.75 D. 1.89

66. ECE Board April 1998  

Arc tan  2 cos      equal to:  A. 3  B. 8  C. 4  D. 6

62. ECE Board March 1996 Solve for x in the equation: Arctan  2x +Arctan  x =





∏ 4

A. 0.218 B. 0.281 C. 0.182 D. 0.896

1   3 2 arcsin     2     







is 

67. ECE Board April 1998 Two electrons have speed of 0.7c and x respectively at an angle of 60.82 degrees between each other. If their relative velocity is 0.65c, find x. A. 0.12c B. 0.16c C. 0.15c D. 0.14c

63. ECE Board April 1998 The side of the triangle are 8, 15, 17 units. If each side is doubled, how many square units will the area of the new triangle be? A. 210 units B. 200 units C. 180 units D. 240 units 64. ECE Board November 1997 Find the 100th tern of the sequence. 1.01, 1.00, 0.99 … A. 0.04 B. 0.03 C. 0.02 D. 0.05

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I D. 41.6 73. Problem: Two sides of a triangle measures 6 cm. and 8 cm, and their included angle is 40Find the third side. A. 5.234 cm B. 4.256 cm. C. 5.144 cm. D. 5.632 cm.

68. ECE Board April 1998 A man finds the angle of elevation of the top of a tower to be 30 degrees. He then walks 85 m nearer the tower and found its angle of elevation to be 60 degrees. What is the height of the tower? A. 95.23 B. 45.01 C. 76.31 D. 73.61

74. Problem: Given a triangle: C = 100, a = 15, b = 20. Find c: A. 27 B. 34 C. 43 D. 35

69. ECE Board November 1998 If an equilateral triangle is circumscribed about a circle of radius 10 cm, determine the side of the triangle. A. 34.64 cm B. 64.12 cm C. 36.44 cm D. 33.51

75. Problem: Given angle A= 32 degree, angle B = 70 degree, and side c = 27 units. Solve for side a of the triangle. A. 10.63 units B. 10 units C. 14.63 units D. 12 units

70. Problem: The angle formed by two curves starting at a point, called the vertex, in a common direction. A. horn angle B. inscribed angle C. dihedral angle D. exterior angle

76. Problem: In triangle ABC, angle C = 70 degrees; angle A = 45 degrees; AB = 40 m. What is the length of the median drawn from the vertex A to side BC? A. 36.8 meters B. 37.4 meters C. 36.3 meters D. 37.1 meters

71. Problem: 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. 16 cm B. 15 cm C. 17 cm D. 18 cm

77. Problem: 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 bisector of angle C. A. 35.53 B. 54.32 C. 52.43 D. 62.54

72. Problem: 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 40kph, how far directly from M, in km. will be it after 2 hours? A. 43.5 B. 47.9 C. 45.2

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I the airplane is 700 miles, when will it lose contact with the carrier? A. 5 meters B. 20 meters C. 10 meters D. 2.13 meters

78. Problem: 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. 20.45 cm B. 22.35 cm C. 21.74 cm D. 20.85 cm.

83. Problem: 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 B. 20 meters C. 10 meters D. 2 5 meters

79. Problem: The sides of a triangular lot are 130m, 180m, and 190m. 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. 115 meters B. 100 meters C. 125 meters D. 130 meters

84. Problem: From the top of the building 100m high, the angle of depression of a point A due East of it is 30o. From a point B due south of the building, the angle of elevation of the top is 60 o. Find the distance AB. 30 A. 100 3 B. 100+3 30

80. Problem: From a point outside of an equilateral triangle, the distance to the vertices is 10m, 10m, and 18m. Find the dimension of the triangle. A. 25.63 B. 19.94 C. 45.68 D. 12.25

C. 100- 3 30 30 D. 100+ 62

81. Problem: Points A and B 1000m 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 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 B. 274 meters C. 284 meters D. 294 meters

85. Problem: An observer found the angle of elevation at the top of the tree to be 27o. After moving 10m closer (on the same vertical and horizontal plane as the tree), the angle of elevation becomes 54o. Find the height of the tree. A. 8.09 meters B. 7.53 meters C. 8.25 meters D. 7.02 meters

82. Problem: An airplane leaves an aircraft carrier and flies South at 350 mph. the carrier travels S 30o E at 25 mph. if the wireless communication range of

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86. Problem: From 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 at an inclination of 30to the horizon, and reaching a point C, an observer finds that the angle ABC is 135. The height of the mountain in feet is: A. 14,986 B. 12,493 C. 14,789 D. 12,225

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 tower. The vertical angle at a point A is 30and at a point B is 40. What is the height of the tower? A. 110.29 feet B. 92.54 feet C. 143.97 feet D. 85.25 feet 91. Problem: Find the supplement of an angle whose compliment is 62 A. 280

87. Problem: A 50 meter vertical tower casts a 62.3 meter shadow when the angle of elevation of the sum is 41.6. the inclination of the ground is: A. 4.33 B. 4.72 C. 5.63 D. 5.17

B. 1520 C. -2 6.20 1 D. 2 92. Problem: A certain angle has a supplement five times its compliment. Find the angle. A. 67.50 B. 157.50 C. 168.50 D. 186.50

88. Problem: A vertical pole is 10m from a building. When the angle of elevation of the sun is 45the pole cast a shadow on the building 1m high. Find the height of the pole. A. 12meters B. 11 meters C. 0 meter D. 13meters

93. Problem: The sum of the 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. 450 B. 750 C. 930 D. 900

89. Problem: A pole cast a shadow of 15 meters long when the angle of elevation of the sun is 61o . if the pole has leaned 15from the vertical directly toward the sun, what is the length of the pole? A. 52.43meters B. 54.23 meters C. 52.25 meters D. 53.24 meters

94. Problem: 1 revolutions 2 counter-clockwise is: A. 5400 B. 5200 C. 5800 D. 5950 The measure of 1

90. Problem: An observer wishes to determine the height of the tower. He takes sights

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95. Problem: The measure of 2.25 revolutions counterclockwise is: A. 800 degrees B. 820 degrees C. 810 degrees D. 850 degrees

100. Problem: The insides of a right triangle are in arithmetic progression whose common difference is 6 cm. Its area is: A. 270 cm2 B. 340 cm2 C. 216 cm2 D. 144 cm2

96. Problem: If Tan  = x2, which of the following is correct? 1 A. cosθ = 1+x4 1 B. sinθ = 1+x4 1 C. cscθ = 1+x4 1 D. tanθ = 1+x4

101. Problem: From the top of tower A, the angle of elevation of the top of the tower B is 46 . From the foot of a 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 120 m., How far is A from the building? A. 42.3 m. B. 40.7 m. C. 38.6 m. D. 44.1 m.

97. Problem: In an isosceles right triangle, the hypotenuse is how much longer than its sides? A. 2 times B. 2 times C. 1.5 tines D. none of these

102. Problem: Point A and B are 100 m apart and are on the same elevation as the foot of the building. The angles of elevation of the top of the building from point A and B are 21 and 32, respectively. How far is A from the building? A. 265.4 m. B. 277.9 m. C. 259.2 m. D. 259.2 m.

98. Problem: Find the angle in mils subtended by a line 10 yards long at a distance of 5,000 yards. A. 1 mil B. 2 mils C. 6 mils D. 3 mils

103. Problem: 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. 73.61 B. 76.31 C. 73.31 D. 71.36

99. Problem: The angle or inclination of ascends of a road having 8.25% grade is _____ degrees. A. 5.12 degrees B. 1.86 degreed C. 4.72 degrees D. 4.27 degrees

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C. 2 2 D. 6 2

104. Problem: The angle of elevation of point C from point B is 2942’; the angle of elevation of C from another point A 31.2 m. directly below is 5923’. How high is C from the horizontal line through A? A. 35.1 meters B. 52.3 meters C. 47.1 meters D. 66.9 meters

108. Problem: 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 6 inches long. The plane 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. 8.23 in. B. 10.65 in. C. 9.17 in. D. 11.25 in.

105. Problem: A rectangle piece of land 40 m x 30 m is to be crossed diagonally by a 10-m wide roadway as shown. If the land cost P1,500.00 per square meter, the cost of the roadway is: A. 401.10 B. 60,165.00 C. 651,500.00 D. 601,650.00

109. Problem: If the bearing of A from B is S 40W, then the bearing of B from A is: A. S 40W B. S 50W C. N 40E D. N 75W

106. Problem: A man improvises a temporary shield from the sun using a triangular piece of wood with dimensions of 1.4 m, 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 0.9 m long. What would be the area of the shadow on the ground when the sun is vertically overhead? A. 0.5m2 B. 0.75m2 C. 0.84m2 D. 0.95 m2

110. Problem: A plane hillside is inclined at an angle of 28 degree with the horizontal. A man wearing skis can climb this hillside by following a straight path inclined at an angle of 12 degree to the horizontal, but one without skis must follow a path inclined at an angle of 5 degree with the horizontal. Find the angle between the directions of the two paths. A. 10.24 degree B. 13.21 degree C. 17.22 degree D. 15.56 degree

107. Problem: A rectangular piece of wood 4 x 12 cm tall is tilted at an angle of 45 degrees. Find the vertical distance between the lower corner and the upper corner. A. 4 2 B. 8 2

111. Problem: The sides of the triangle ABC are AB = 15 cm, BC = 18 cm, CA = 24 cm. Find the distance from the point of intersection of the angle bisectors to side AB.

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A. 5.45 B. 5.34 C. 4.73 D. 6.25 112. Problem: Two straight roads intersect to form an angle of 75 degrees. Find the shortest distance from one road to a gas station on the other road 1 km., from the junction. A. 1.241 B. 4.732 C. 2.241 D. 3.732

A. 45,065,746.09 B. 56,476,062.07 C. 64,754,034.02 D. 24,245,258.00 117. Problem: If the Greenwich Mean Time (GMT) is 7 A.M. What is the time in a place located at 135 degrees east longitude? A. 4 P.M. B. 6 P.M. C. 2 P.M. D. 5 P.M. 118. Problem: If Greenwich Mean Time (GMT) is 9 a.m. what is the time in a place 45o W of longitude? A. 6 A.M. B. 4 A.M. C. 2 A.M. D. 8 A.M.

113. Problem: A train travels 2.5 miles up on a straight track with a grade of 110’. What is the vertical rise of the train in that distance? A. 0.716 miles B. 0.051 miles C. 0.279 miles D. 0.045 miles

119. Problem: Find the distance in nautical miles and the time difference between Tokyo and Manila if the geographical coordinates of Tokyo and Manila are ( 35.65 degree north Lat.; 139. 75 degrees East long.) and ( 14.58 degree North; 120. 98 degree long.), respectively. A. 1469 nautical miles; 4.25 hrs. B. 2615 nautical miles; 1.52 hrs. C. 1612 nautical miles; 1.25 hrs. D. 1485 nautical miles; 1.25 hrs.

114. Problem: Four holes are to be spaced regularly on a circle of radius 20 cm. Find the distance “d” between the centers of the two successive holes. A. 20 2 B. 10 3 C. 15 2 D. 5 2 115. Problem: In a spherical triangle ABC, A = 11619’, B = 5530’ and C = 8037’. Find the value of side a. A. 115.57 degree B. 110.56 degree C. 118.17 degree D. 112.12 degree

120. Problem: An isosceles spherical triangle has angle A=B= 54 degree and side b= 82 degrees. Find the measure of the third angle. A. 15824’15” B. 15524’15” C. 16824’15” D. 16524’15”

116. Problem: Considering the earth as a sphere of radius 6,400 km. Find the area of a spherical triangle on the surface of the of the earth whose angels are 5089’ and 120.

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I An observer 9 m. horizontally away from the tower observes its angle of elevation to be only one half as much as the angle of elevation of the same tower when he moves 5 m. nearer towards the tower. How high is the tower? A. 6 m B. 4 m C. 3 m D. 5 m

121. Problem: On one side of a paved oath walk is a flag staff on top of it. The pedestal is 2 m. in height whole the flag staff is 3 m, high. At the opposite edge of the path walk the pedestal and flag staff subtends equal angles. Compute the width of the path walk. A. 4.47 m B. 5.34 m C. 5.74 m D. 3.78 m

126. Problem: 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 elevations to be 60 degrees. What is the height of the tower? A. 73.61 m B. 45.36 m C. 66.36 m D. 54.21 m

122. Problem: Simplify the equation sin2  (1 + cot2 ) A. Sin 2  B. 1 C. Sin 2  Sec 2  D. Cos 2 

127. Problem: Points A and B are 100 m. Apart and are of the same elevation as the foot of the bldg. The angles of elevation of the top of the bldg. from points A and B are 21 degrees and 32 degrees respectively. How Far is A from the bldg. in meters? A. 277.36 B. 271.62 C. 265.42 D. 259.28

123. Problem: The angle of elevation of a top of a tree from a point 10 m. horizontally away from the tree is twice the angle of elevation at a point 50 m. from it. Find the height of the tree. A. 34.25 B. 27.89 C. 46.58 D. 38.73

128. Problem: A and B are summits of two mountains rise from a horizontal plain. B being 1200 m above the plain. Find the height of A, it being given that its angle of elevation as seen from a point C in the plane (in the same vertical plane with A and B) is 50 degree, while the angle of depression of C viewed from B is 2858’ and the angle subtended at B by AC is 50 degree. A. 3200.20 m B. 3002.33 m C. 2989.42 m

124. Problem: A vertical pole consists of two parts, each one half of the whole pole. At a point in the horizontal plane which passes through the foot of the pole and 36 m. From it, the upper half of the pole subtends an angle whose tangent is 1 / 3. How high is the pole? A. 74 m or 36 m B. 72 m and 36 m C. 60 m or 30 m D. 80 m or 40 m 125. Problem:

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D. 2847.64 m

Four hours earlier a freight ship started from the same point at the speed of 8 kph with a direction N 1542’ W. Determine the number of hours it will take the freight ship to be exactly N 7525’ W of the passenger ship. A. 2 B. 4 C. 8 D. 5

129. Problem: A 40 m high tower stands vertically on a hillside (sloping ground) which makes an angle of 18 degree with the horizontal. A tree also stands vertically up the hill from the tower. An observer on the top of the tower finds the angle of depression of the top of the tree to be 26 degrees and the bottom of the tree to be 38 degree. Find the height of the tree. A. 15.29 m B. 10.62 m C. 7.38 m D. 13.27 m

133. Problem: A truck travels from point M north ward for 30 min., then eastward for one hour, then shifted N. 30 degrees West. If the constant speed is 40 kph, how far directly from M in km, will it be after 2 hours? A. 45.22 B. 47.88 C. 41.66 D. 43.55

130. Problem: Two towers A and B stands 42, apart on a horizontal plane. A man standing successively at their bases observes that the angle of elevation of the top of tower B is twice that of the bases the angles of elevation are complimentary. Find the angle of elevation of the tower B from the base of tower A if the height of the tower B is 30 m. A. 28.32 degree B. 38.58 degree C. 40.27 degree D. 38.58 degree

134. Problem: A car travels northward from a point B for one hour, then eastward for 30 one hour, then eastward for 30 min then shifted N 30 E. After exactly 2 hours, the car will be 64.7 km directly away from B. What is the speed of the car in Kph? A. 45 B. 40 C. 50 D. 55

131. Problem: A ship started sailing S 4235’ W at the rate of 5 kph. 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. 4.03 B. 3.84 C. 2.96 D. 5.8

135. Problem: A motorcycle travels northward from point L for half an hour, then eastward for one hour, then shifted N 30 W. After exactly 2 hours, the motorcycle will be 47.88 km. Away from L. What is the speed of the motorcycle in kph? A. 45 B. 43 C. 130 D. 33

132. Problem: A passenger ship sailed northward with a direction N 4225’ E at 14 kph.

136. Problem:

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A boat can travel 8 mi / hr. in still water. What is its velocity with respect to the shore if it heads 55 degrees east of north in a current that moves 3 mi/hr west? A. 5.4 mph B. 6.743 mph C. 4.556 mph D. 8.963 mph

The area of an isosceles triangle is 36 m2 with the smallest angle equal to one third of the other angle. Find the length of the shortest side. A. 5.73 m. B. 9.22 m. C. 8.46 m. D. 12.88m. 142. Problem: The difference between the angles at the base of a triangle is 17 48’ and the sides subtending this angles are 105.25 m and 76.75 m. Find the angle included between the given sides. A. 90 degrees B. 80 degrees C. 70 degrees D. 60 degrees

137. Problem: If the figure, BD and DC are angle bisectors. If angle A= 80, how many degrees is angle ADC? A. 140 degrees B. 120 degrees C. 150 degrees D. 130 degrees 138. Problem: if a triangle ABC, angle A = 60 degrees and the angle B=45 degrees, what is the ratio of sides BC to side AC? A. 1.36 B. 1.22 C. 1.48 D. 1.19

143. Problem: Three forces 20 N, 30 N and 40 N are in equilibrium. Find the angle between the 30 N and 40 N forces. A. 25.97 degrees B.40 degrees C. 28.96 degrees D. 3015’25”

139. Problem: Find the angle B of a triangle if a = 132 m., b = 224 m. And C = 28.7 degrees. A. 5903’25” B. 4903’25” C. 3903’25” D. 1903’25”

144. Problem: One leg of the right triangle is 20 units and the hypotenuse is 10 units longer than the other leg. Find the lengths of the hypotenuse. A. 20 B. 25 C. 10 D. 15

140. Problem: The area of an isosceles triangle is 72 sq.m. If the two equal sides make an angle of 20 degree with the third side, compute the length of the longest side. A. 25.62 B. 27.84 C. 31.22 D. 28.13

145. Problem: Determine the sum of the positive valued solution to the simultaneous equations: xy = 15, yz = 35, zx = 21. A. 15 B. 13 C. 17 D. 19

141. Problem:

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I the sum is 20 degrees and its bearing is S 60E, calculate the area of the shadow of the wall on the horizontal ground. CD is on the ground portion of the wall and has a direction of due north. A. 78.53 sq. m. B. 80.30 sq. m. C. 73.45 sq. m. D. 71.40 sq. m.

146. Problem: A rectangle ABCD which measures 18 x 24 units is folded once, perpendicular to diagonal AC, so that the opposite vertices A and C coincide. Find the length of the fold. A. 18.75 cm. B. 22.5 cm. C. 21.5 cm. D. 19.5 cm.

151. Problem: A spherical triangle ABC has an angle C = 90 degrees and sides a = 50 degrees and c = 80 degrees. Find the value of “b” in degrees? A. 75.44 B. 74.33 C. 76.55 D. 73.22

147. Problem: ABDE is a square section and BDC is an equilateral triangle with C outside the square. Compute the value of angle ACE. A. 35 degree B. 50 degree C. 30 degree D. 40 degree

152. Problem: A point O is inside a square lot, if the distances from point O to the three successive corners of the square lot are 5 m., 3 m. And 4 m, respectively, find the area of the square lot. A. 32.1 B. 23.2 C. 45.4 D. 36.6

148. Problem: Find the sum of the interior angles of the vertices of a five pointed star inscribed in a circle. A. 120 degrees B. 140 degrees C. 180 degrees D. 170 degrees

153. Problem: Find the angle in mils subtended by a line 10 yards long at a distance of 5,000 yards. A. 1.0 yards B. 2.04 mils C. 1.5 yards D. 1.3 yards

149. Problem: A square section ABCD has one of its sides equal to x. Point E is inside the square forming an equilateral triangle BEC having one side equal to the side of a square. It is required to compute the angle of AED. A. 150 degrees B. 140 degrees C. 150 degrees D. 120 degrees

154. Problem: The A point F is inside an equilateral triangle, if the distances from F to the three vertices of the triangle are 3 m, 4 m, and 5 m, respectively, Find the area of the triangle A. 17.5 B. 12.23 C. 19.85 D. 20.0

150. Problem: ABCD is a vertical wall AD = 3 m. High, AB = 10 m. long. The wall is built on a north south line on a horizontal ground. If the elevation of

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I The corners of a triangle lot are marked 1, 2, and 3 respectively. The length of side 3 – 1 is equal to 500 m. The angles 1, 2 and 3 are 60 degrees, 80 degrees and 40 degrees respectively. If the area of 59,352 sq. m. is cut off on the side 3 – 1 such that the dividing line 4 -5 is parallel to 3 – 1. 1. Compute the length of line 4 -5. 2. Compute the area of 2 – 4 – 5. 3. Compute the distance 2 – 4.

155. Problem: If the sides of the triangle are 2x + 3, x2 + 2x, find the greatest angle. A. 120 degrees B. 100 degrees C. 110 degrees D. 130 degrees 156. Problem: Find the value of  in the equation Cosh 2 x – Sinh 2  =2cos. A. 45 degrees B. 60 degrees C. 30 degrees D. 35.6 degrees

161. Problem: What are the exact values of the cosine and tangent trigonometric functions of acute angle A, Given 3 that Sin A = . 7

157. Problem: Solve for x if Cosh x + Sinh x = 7.389 A. 2 B. 10 C. 1 D. ex

A. B. C.

158. Problem: Find x if Cosh 2 x – Sinh 2 x + tanh 2 x + sech 2 x= Cosh x + Sinh x A. 1 B. ln 2 C. ex D. x-x

D.

2 8 5

3 10 20 4 10 15 3 8 10

162. Problem: Simplify the expression sec  - (sec ) sin 2  A. Cos 2  B. Cos  C. Sin  D. Sin 2 

159. Problem: Triangle XYZ has base angles X = 52 degrees and Z = 60 degrees distance XZ = 400 m long. A line AB which is 20 m long is laid out parallel to XZ. 1. Compute the area of triangle XYZ. 2. Compute the area of ABXZ. 3. The area of ABY is to be divided into two equal parts. Compute the length of the dividing line which is parallel to AB.

163. Problem: A flagpole is places on top of the pedestal at a distance of 15 m from the observer. The height of the pedestal is 20m. If the angle subtended by the flagpole at the observer is 10 degrees. 1. Compute the angle of elevation of the flagpole. 2. Compute the height of the flagpole.

160. Problem:

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3. If the observer moves a distance of 5m. towards the pedestal, what would be the angle of pedestal, what would be the angle of elevation of the flagpole at this pt.

C. 2 mils D. 7 mils 169. Problem: In an isosceles right triangle, the hypotenuse is how much longer than its sides? A. 2 times B. 2 times C. 1.5 times D. none of these

164. Problem: A right spherical triangle has an angle C = 90 degrees, a = 50 degrees, and c = 80 degrees. Find the side b. A. 78.66 degrees B. 45.33 degrees C. 75.89 degrees D. 74.33 degrees

170. Problem: If tan  = x2, which of the following is incorrect? 1 A. sec  = 1  X4 1 B. cos  = 1  X4 1 C. sin  = 1  X4 1 D. cot  = 1  X4

165. Problem: Calculate the area of a spherical triangle whose radius is 5 m and whose angles are 40 degrees, 65 degrees, and 110 degrees. A. 15. 27 B. 17.23 C. 21.21 D. 15.87 166. Problem: The sides of right triangle are in arithmetic progression whose common difference if 6 cm. Its area is: A. 270 cm2 B. 216 cm2 C. 140 cm2 D. 160 cm2

171. Problem: Find the value of y: y = (1 + cos 2) tan . A. sin  B. sin 2 C. cos 0 D. tan  172. Problem:

167. Problem: The angle of inclination of ascends of a road having 8.25 % grade is ____ degrees. A. 4.72 B. 5.12 C. 4.27 D. 1.86

Simplifying the equation sin 2  (1 + cot 2 ) gives: A. 0.5 B. eg C. 1 D. e2X 173. Problem: Which of the following expression in equivalent to sin 2 A. 2tancot B. 2sincos C. 2sin D. cot

168. Problem: Find the angle in mils subtended by a line 10 yards long at a distance of 5,000 yards. A. 4 mils B. 2.5 mils

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I 180. Problem: The ____ is the line or line segment that divides the angle into two equal parts. A. angle bisector B. apothem C. perpendicular bisector D. terminal side

174. Problem: The equation 2 sinh x cosh x is equal to: A. e-x B. ex C. cosh x D. sinh 2x 175. Problem: Find the value of sin (90+A) A. cos A sin A. B. – cos A C. cos A D. –sin A

181. Problem: _____ are the lines bisecting the angles formed by the sides of the triangles and their extensions. A. Exsecants B. Internal angle bisector C. Perpendicular bisector D. Exterior angle bisectors

176. Problem: If sin(x + y) = 0.766 and sin (x – y) =0.1736. Find sin x cos y. A. 0.9695 B. 0.6732 C. 0.4698 D. 0.8563

182. Problem: The two legs of the triangle are 300 and 150 each, respectively. The angle opposite the 150 m side is 26 degree. What is the third side? A. 341.78 m B. 218.61 m C. 282.15 m D. 175.23 m

177. Problem: Solve for  if coth2x – csch2 x = exsecant . A. 45 B. 30 C. 60 D. 20

183. Problem: The sides of the triangular lot are 130, 180 and 180 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 B. 240 C. 125 D. 200

178. Problem: A transformation consisting of a constant offset with no rotation or distortion. A. screw B. translation C. reflection D. torsion

184. Problem: The sides of the triangle are 195, 157 and 210, respectively. What is the area of the triangle? A. 14, 586.2 B. 28, 586 C. 16, 586.2 D. 41, 586.2

179. Problem: An angle whose endpoints are located on a circle’s circumference and vertex located at the circle’s center A. central angle B. exterior angle C. supplementary angle D. complementary angle

185. Problem:

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If sin A =

I The turning of an object or coordinate system by an angle about a fixed point. A. involution B. revolution C. dilation D. rotation

3 and sin  A +B  =1, find 5

cos B: A. 0.70 B. 0.80 C. 0.60 D. 0.100

186. Problem: A Meralco 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 Meralco tower at 13 degrees and 35 degrees respectively. The height of the tower is 50 m. Find the height of the monument. A. 33.541 m B. 64.12 m C. 32.10 D. 36.44 m

190. Problem: A rotation combined with an expansion or geometric con traction. A. screw B. shift C. twirl D. twist 191. Problem: Find the sin x if 2sinx + 3cosx - 2 = 0 A. 1 and 5 / 13 B. 3 and 5 / 15 C. 15 and 5 / 13 D. 1 and -5 / 13

187. Problem: A wire supporting a pole is fastened to it 20 feet from the ground and to the ground 15 feet from the pole. Determine the length of the wire and the angle it makes with the pole. A. 25 ft, 36.87 degrees B. 25 ft, 53.17 degrees C. 24 ft, 36.87 degrees D. 24 ft, 53.17 degrees

192. Problem: If sin A = 4 / 5, A in quadrant II, sin B = 7 / 25, B in quadrant I, find the sin (A + B) A. 2 / 5 B. 3 / 5 C. 4 / 5 D. 6 / 5

188. Problem: Point 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 degrees and 32 degrees respectively. How far is A from the building in meters? A. 259.28 B. 325.00 C. 454.85 D. 512.05

193. Problem: If sin A = 2.57 x; cos A = 3.06x, and sin 2A = 3.939x, find the value of the x. A. 0.150 B. 0.100 C. 0.250 D. 0.350 194. Problem: if cos  = 3 2 , then find the value

189. Problem:

of x if x = 1- tan 2 ? A. 2 / 3

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B. 4 / 3 C. 8 / 9 D. 1 / 9 195. Problem: If sin  - cos  = - 1/3, what is the value of sin 2? A. 8 / 9 B. 4 / 9 C. 10/ 12 D. 10 / 15

A. 4114’48”, 6310”48” B. 4523’43”, 6612’45” C. 4013’35”, 6612’45” D. 2224’3”, 6012’45”

196. Problem: Given the parts of the spherical triangle: A = 6030’ b = 3815’ a = 4030’

198. Problem: From the given parts of the spherical triangle ABC, compute for the angle A. A = 5230’ B = 4834’ C = 12015’

A. 4535’ B. 4430’ C. 4743’ D. 4640’ 197. Problem: In the spherical triangle shown, following parts are given: A = 4018’ C = 7500’ c = 10010’ b = 6525’

A. 12842’ B. 5716’ C. 14132’ D. 11416’ 199. Problem: The ____ is an imaginary rotating sphere of gigantic radius with the earth located at its center. A. celestial sphere B. chronosphere C. exosphere D. astrological sphere 200. Problem:

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Two celestial coordinate are: A. right ascension and declination B. longitude and latitude C. North Pole and South Pole D. zenith and nadir

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