Trigo.docx

1. If csc2 θ = x2 + 1, what is cot2 θ? Ans: x2 2. What is the value of x in Arctan 3x + Arctan 2x= 45˚ Ans: 1/6 3. A triangular fish pen has sides 30 cm, 50cm and 60cm. Find the acute angle opposite to the shortest side. Ans: 30˚ 4. Given the curve y=3cos1/2x. Find the amplitude and period. Ans: 3, 4pi 5. Simplify the equation sin2 θ (1+ cot2 θ) Ans: 1 6. If cos z=2, find cos 2z. Ans: 7 7. Simplify each expression using the fundamental identities. Sin2 x + Cos2 x Tan x Ans: Cot x 8. Simplify (cos 0˚ + cos 1˚ + cos 2˚ + … + cos 90˚)/(sin 0˚ + sin 1˚ + sin 2˚ + … + sin 90˚) Ans: 1 9. Solve for G is csc(11G - 16 degrees) = sec (5G + 26 degrees). Ans: 5˚ 10. If sin(x) = 2/5 and x is an acute angle, find the exact value of cos(2x) Ans: 17/25 11. Add 1 + 1 Sinθ Cosθ Ans: cosθ + sinθ sinθ cosθ 12. Find the value of sin (arc cos 15 /17) Ans: 8/17 13. If conversed sin θ = 0.134, find the value of θ. Ans: 60˚

the antenna, signing along the guy wire, is 42.3˚, then what is the height of the antenna? Ans: 72.7m 17. 10. A photographer wants to take a picture of a 4 feet vase standing on a 3 feet pedestal. She wants to position the camera at point c on the floor so that the angles subtended by the vase and the pedestal are the same size. How far away from the foot of the pedestal should the camera be? Ans: 7.9ft 18. Two cities are 270 miles apart lie on the same meridian. What is the difference in latitude, if the radius of the earth is 3,960 miles? Ans: 3/44 rad 19. Two stones are 1 mile apart and are of the same level as the foot of the hill. The angles of depression of the two stones viewed from the top of the hill are 5 degrees and 15 degrees respectively. Find the height of the hill. Ans: 209.01m 20. A ship leaves port and travels 36 miles due west. It then changes course and sails 24 miles on a bearing of S 33˚W. How far is it from port at this point? Ans: 53 miles 21. An engineer is hired to design a bridge to carry traffic across the river. Her first problem is to determine the distance from point A to point B. To do so,she starts at point A and measures distance of 250m, in a direction at right angles to the segment 49˚50’. How long should the bridge be? Ans: 296. 184m

15. Find the height of a tree if the angle of elevation of its top changes from 20˚ to 40˚ as the observer advances 75 ft. toward its base. Ans: 48. 21 m

22. Two cars A and B started at the same time form the same point and moved along straight line which intersects at an angle of 60˚. If car A was moving at the rate of 50 kph and car B at the rate of 70 kph, how far apart are they at the end of 45 minutes? Ans: 46.84 km

16. A guy wire of length 108 meters runs from the top of an antenna to the ground. If the angle of elevation of the top of

23. AB and CD are two buildings. The angle of elevation of A from C and D are θ and 45˚ respectively.

14. Find the exact value of [ tan (25°)+ tan (50° ] / [ 1 - tan( 25°) tan(50°) ]. Ans: √(3) + 2

Find the height. Ans: h/(1- tan θ) 24. From a ship sailing due east to a lighthouse was seen to bear N 45˚ E. After sailing 5 km the lighthouse has a bearing of N 69˚ W. How far was the lighthouse from the both points of observation ? Ans: 3.87 km 25. A hiker walk 8000m on a course of S 81˚ E. She then changes direction and hikes 5000m on a course of N 32˚W. How far is she from her starting point, and on what course must she travel to return to the starting point? Ans: 6042.8 m; S 60.4˚ W 26. 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, 9.75 m above the first point of observation the angle of depression is 58˚35’. How far is the object from the building. Ans: 10.7 m 27. The sides of the triangle are 17, 21 , and 28 m respectively. Determine the length of the line bisecting the greatest side and drawn from the opposite angle. Ans: 13 m 28. An aerolift airplane can fly at an airspeed of 300mph. If there is a wind blowing towards the cast at 50 mph. 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 the course. Ans: 21.7 ˚ , 321.8 mph 29. 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 the tower. The vertical angle at point A is 30˚ and at point B is 40˚. What is the height of the tower ? Ans: 92.54 ft. 30. The angle of elevation of the top of tower B from the top of tower A is 28˚ and the angle of elevation of the top of tower A from the base of tower B is 46˚. The towers lie on the same lie in the same horizontal plane. If the height of tower B is 120m, find the height of tower A. Ans: 79.3 m 31. An observer finds the angle of elevation of a building from X on a level ground and as 27˚. After moving 120 m closer to the building the angle of elevation was 42˚. Find the height of the building. Ans:140.85 m 32. A flagpole 3m high stands at a top of a pedestal 2m high located at one side of a pathway. At the opposite side of the pathway directly facing the flagpole, the flagpole subtends the same angle as the

pedestal. What is the width of the pathway? Ans: 4.47 m 33. From a window of a building 4.25 m above the ground, the angle elevation of the top of nearby building is 36.6˚ and the angle of depression of its base is 26.2˚. What is the height of the nearby building? Ans: 10.665 m 34. 168, 187,215, respectively. What is the area of the triangle ? Ans:15124.37 sq. units 35. The sides of a triangle are 8, 15 and 17. If each side is doubled, how many square units will the area of the new triangle be? Ans: 240 36. The two legs of a triangle are 300 and 150m each, respectively. The angle opposite the 150m side is 26˚.What is the third side ? Ans: 341. 78m 37. If an equilateral triangle is circumscribed about a circle of radius 10 cm, determine the side of the triangle. Ans: 34.64 cm 38. The triangle ABC has a side AB=160 cm, BC= 190 cm, and CA= 190 cm. Point D is along side AB and AD= 100 cm. Point E is along side CA. Determine the length of AE if the area of a triangle ADE is 3/5 the area of triangle ABC. Ans:182.4 cm 39. An observer wishes to determine the height of the tower. He observed the top of the tower from A and got an angle of elevation 30˚. He then walked 25m closer to point B and observed the angle of elevation as 40˚ Points A and are the same elevation, and on a direct line with the tower.How high is the tower ? Ans: 46.27 m 40. 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. Ans: 25 ft, 36.87˚ 41. If Greenwich Mean Time (GMT) is 6 A.M., what is the time at a place located 30˚ East longitude ? Ans: 8 A.M. 42. If the longitude of Tokyo is 139˚E and that of Manila is

121˚E, what is the time difference between Tokyo and Manila? Ans: 1hr and 12min. 43. One degree on the equator of the earth is equivalent to ____ in time. Ans: 4 minutes 44. A spherical triangle ABC has an angle C = 90˚ and sides a= 50, and c= 80˚. Find the value of “b” in degrees. Ans: 74.33 45. Solve the remaining side of the spherical triangle whose given parts A= B = 80˚ and a=b= 89˚ Ans: 168˚31’ 46. Solve for side b of a right spherical triangle ABC whose parts are a= 46˚, c= 75˚ and C= 90˚. Ans: 68˚ 47. Given a right spherical triangle whose parts are a= 82˚, b=62˚ and C=90˚. What is the value of the side opposite the right angle ? Ans: 86˚15’ 48. Solve the angle A in the spherical triangle ABC, given a= 106˚25’, c= 42˚16’ and B= 114˚53’. Ans:97˚09’ 49. Determine the spherical excess of the spherical triangle ABC given a= 56˚ , b= 65˚ and c= 78˚. Ans: 33˚33’ 50. What is the spherical excess of a spherical triangle whose angles are all right angles? Ans:90˚ 51. The area of spherical triangle ABC whose parts are A= 93˚40’, B=64˚12’, C= 116˚51’ and the radius of the sphere is 100m is: Ans: 16531 sq. m. 52. A spherical triangle has an area of 327.25 sq. Km. What is the radius of the sphere if its spherical excess is 30˚ Ans:25 km 53. The sides of a triangle lot are 130 m, 180m, and 190m. 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. Ans: 125 m

54. Determine the area of a spherical triangle ABC if A= 140˚, B = 75˚ and C= 86˚, if it has a radius of 40 km. Ans: 3379 sq. Km. 55. A lighthouse is 10 units of length northwest of a dock at 8:00 am and travels west at 12 units of length per hour. At what time will the ship be 8 units of length from the lighthouse? Ans: 8:54.1 AM 56. Find the value of θ if conversed sin θ= 0.256855. Ans: 48˚ 57. Two inaccessible objects A and B are each viewed from two stations C and D on the same side of AB and 562 m. apart. The angle ABC is 62˚12’, BCD= 41˚08’. ADB= 60˚49’ and ADC is 34˚51’. Find the required distance AB. Ans: 739.90 m 58. The area of triangle ABC is 45 sq. Units. If a= 10 & b= 15, find the measure of angle C to the nearest degree Ans: 143.1˚ 59. In triangle ABC, tan A + tan B + tan C =4. Find the value of tan A tan B tan C. Ans: 4 60.  A balloon is hovering 800 ft above a lake. The balloon is observed by the crew of a boat as they look upwards at an angle of 0f 20 degrees. 25 seconds later, the crew had to look at an angle of 65 degrees to see the balloon. How fast was the boat traveling? Ans: 73ft/s 61.Solve for x from the following equation: cos 6 x=¿

1 ¿ csc ⁡(3 x +9)

Note: all units are in degrees. Answer: x=9 ° 62. If csc ( A+ x )=sec ( B+2 y), then A+B is: Answer: A+ B=90 °−(x +2 y) 63. Find the value of ϕ in the equation

cosh 2 ϕ−sinh2 ϕ=3 sinϕ Answer: ϕ=19.47 ° 64.sin x=3 /5 , tan y =2/5. Find sin( x + y ) Answer: sin( x + y )=0.854 65. Find the value of x such that the tangent of angle (2 x+18) is equal to the cotangent of the angle( 4 x−12) . Answer: x=14 ° 66. If arcsin (6x + 4y) = 1.5708 and arccos ( 2 x + y )=1.0472, find x. Answer:

x=

1 2

67. What is the angle between the hands

of a clock at 3:25 pm? Answer: 47.5 ° 68. Given a triangle ABC such that

A=40 ° , a=10 m ,∧b=15 m. 69. How many possible triangles can be formed? Answer: 2 69. If a=4 cos x+ 6 sin x and b=6 sin x−4 cos x , what is the value of a 2+b 2? Answer: a 2+b 2=32+40 sin 2 x 70. A 6-m ladder leans against a vertical wall. If the top of the ladder is 4.8m above the floor, how far is its foot from the wall? Answer: 3.6 m 71. A surveyor is 1190 m and 2290 m away from the two edges of a lake. If the angle between measurements is 71 °, what is the width of the lake? Answer: 2,210.38 m 72. The sides of a triangular lot are 28.45 m, 47.12 m and 35.64 m. What is the angle opposite the 47.12 side? Answer: 93.978 ° 73. If sin θ=2/3∧cos θ< 0, find tanθ . Answer: tanθ=−2 √ 5 /5 74. If cosine θ=0.344 , what is the value of θ ? Answer: θ=41 ° 75. Solve for side b of a right spherical triangle ABC whose parts are a=46 ° , c=75 °∧c =90 °. Answer: 68.12 ° 76. Three forces of 20, 30, 40 pounds, respectively, are in equilibrium. Find the angles that they make with each other. Answer: A=28 ° 57'

B=104 ° 29' C=46 ° 34 '

77. sin x=0.5. Find cos 2 x . Answer: cos 2 x=0.5 78. An airplane having a speed of 120 miles an hour in calm air is pointed in a direction 30 ° east of north. A wind having a velocity of 15 miles an hour is blowing from the northwest. Find the speed and direction of the airplane relative to the ground. Answer: 117 miles/hour ;

37 ° 7 ' east of north

79. A spherical triangle have angles 63 ° ,85 ° ∧54 ° . Find the area of the triangle if the sphere has a surface area of 16,286 square centimeters. Answer: A=497.63 cm2 80-82. Situation 1: The second angle of a triangle is twice the first angle and the third angle is thrice the second angle. The perimeter of the triangle is 250cm. 80. What is the value of the second angle of the triangle? Answer: B=40 ° 81. What is the length of the longest side of the triangle? Answer: 116.98 cm 82. What is the length of the angle bisector to the longest side of the triangle? Answer: 30.15 cm 83-85. Situation 2: Solve the following: 83.

tan

( 116π )=¿ ¿

Answer: −√ 3 /3 84.

tan

( −76 π )=¿ ¿

Answer: −√ 3 /3

elevation of a cliff is 12 °. If the height of the cliff above the sea level is 30 m, how far is he from the shore? Answer: 141.14 m 91-93. Situation 3: Points A and B on opposite sides of a building are 124 m apart. The angles of depression from the top of the building to A and B are 54.3° and 68.5° respectively. Determine the following: 91. The height of the building. Answer: 111.46 m 92. Distance of point A from the building. Answer: 80.09 m 93. Distance of point B from the building. Answer: 43.91 m 94. A spherical triangle ABC,

A=116 ° 19 ' , B=55 ° 30 ' ∧C=80 ° 37 '

. What is the value of side b? Answer: 56 ° 28'

95. A triangle on the surface of sphere has an interior angles of 82 ° ,68 °∧105 ° . The volume of the sphere is 145,124.72 cc. What is the area of the triangle in square cm.? Answer: A=1,391.15 cm2 96-98. Situation 4: In triangle ABC, side AB is 54 cm, side AC is 78 cm, and angle A is 56 ° . 96. What is the measure of side BC? Answer: 65.49 cm

85. tan π =¿ Answer: 0

97. What is the measure of angle B? Answer: 80.88 °

86. For an acute angle θ , cos θ=4 /5. What is the value of cos 2 θ ?

98. What is the measure of angle C? Answer: 43.12 °

Answer: cos 2 θ=

7 25

87. Two flies flew at the same time from the same point in different directions. One fly flew at 0.7 m/s and the other fly at 0.5 m/s. If the angle between their paths is 85 ° , find their distance after 2.5 seconds. Answer: 2.06 m 88. A right triangle ABC has its right angle at B. If AC=x + y and B=x− y , what is the value of side BC? Answer: 2 √ xy 89. If the perpendicular bisector of two sides of a triangle meets at the third side, what kind of triangle is being described? Answer: Right Triangle 90. A fisherman at sea observed that the angle of

99. The perimeter of a triangle is 271 cm. The interior angles measure 50 ° ,60 ° .∧70 ° respectively. What is the length of the longest side of the triangle? Answer: 99.02 cm 100. Simplify the expression: Answer: x=cos θ

sec θ−¿ ¿

101. A triangular lot ABC has the following data:

Angle A=40 ° , Angle C=60 ° , ¿ side AC =24 m. Find the length of side BC? Answer: 15.665 m 102-104. Situation 5: From a window “O” of a tall building 92 meters above a level ground, the angle of depression of the top of a nearby tower is 38 ° . From the base of the building, the angle of elevation of the top of the tower is 22 °. Determine the following: 102. The height of the tower in meters. Answer: 31.36 m 103. The distance from the tower to the

building in meters. Answer: 77.62 m 104. The angle subtended by the tower from point O in degrees. Answer: 11.85 ° 105. A ladder 12 m long with its foot on the street rest against the wall of a house on one side of the street 10 m high. By moving the ladder about its foot, it can reach the wall of the house on the opposite side of the street, but 3 m back of the street line, at a point 6 m high. Find the width of the street. Answer: 14.02 m 106. 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? Answer: 47.88 km 107-109.Situation 6: Which of the following is equivalent to the given trigonometric expressions? 107. ¿ Answer: cos x /( cos2 x−1) 108. ( 1/sin x )( cot x ) Answer: csc 2 x /sec x 2

109. sec x−1 Answer: cot 2 x 110. If cos 65 ° +cos 55 °=cos θ , find θ in radians. Answer: θ=0.087 rad 111.¿. What quadrant does A terminates? Answer: Quadrants I & IV 112-114. Situation 7: From the building across the angle of depression of the base of the front edifice is 23.5 ° and the angle of elevation of its top is 52.78 °. The height of observation is 14 m. 112. What is the height of the edifice? Answer: 56.388 m 113. How far is the building from the said edifice? Answer: 32.198 m 114. What is the angle of elevation of the top of the edifice from the foot of the building? Answer: 60.274 ° 115. The length of the sides of a triangle are AB=20 cm, BC= 24 cm and AC= 17 cm. What is the angle opposite to side BC?

Answer: 80.43 ° 116. The three sides of a triangle measure 36 cm, 18 cm and 24 cm. What is the length of median drawn the longest side to opposite vertex? Answer: 11.225 cm 117-119. Situation 8: The angle of elevation of the top point D of a tower from A is 24.35 °. From another point B the angle of elevation of the top of the tower is 56.21° . The points A and B are 287.6 m apart on the same horizontal plane as the foot (point C) of the tower. The horizontal angle subtended by A and B at the foot of the tower is 90 ° . Determine the following: 117. The distance AC in meters. Answer: 275.27 m 118. The distance BC in meters. Answer: 83.37 m 119. The height of the tower in meters. Answer: 124.58 m 120. A flagpole stands on the edge of the top of a building. At point 200 m from the building the angles of elevation of the top and bottom of the pole are 32 °∧30 ° respectively. Calculate the height of the flagpole. Answer: 9.50 m 121. A person 100 meters from the base of a tree, observe the angle between the ground and the top of the tree is 18 degrees. Estimate the length of the tree to the nearest tenth. Answer: 32.50m 122. The angle of elevation of a hot air balloon, climbing vertically changes from 25 degrees at 10:00 am to 60 degrees at 10:02 am. The point of observation of the angle is situated 300 meters away from the take off point. Find the constant upward speed neglecting air resistance. Answer: 3.16m/s 123. An airplane is approaching point A along a straight line and at a constant altitude ‘h’ at 10:00 am, the angle if elevation of

the airplane is 20 degrees, and at 10:01 am it is 60 degrees. What is the altitude of the airplane if the speed is constant at 600 miles/hour? Answer: 4.60miles 124. When the top of the mountain is viewed from A, 200m from the ground, the angle of depression is equal to 15 degrees and when viewed from point B on the ground, the angle of elevation is 10 degrees. If points A and B are on a same vertical plane. Find the length of the mountain. Answer: 793.80m 125. The area of a right triangle is 50. One of its angles is 45°. Find the lengths of the sides and hypotenuse of the triangle. Answer: 14.14 126 In a right triangle ABC, tan(A) = 3/4. Find the hypotenuse. Answer: 5 127.In a right triangle ABC with an area 60. One of its angles is 45°. Find the lengths of the sides Answer: 10.95, 10.95 128. A rectangle has dimensions 10 cm by 5 cm. Determine the measures of the angles at the point where the diagonals intersect. Answer: 53degrees 129. The lengths of side AB and side BC of a scalene triangle ABC are 12 cm and 8 cm respectively. The size of angle C is 59°. Find the length of side AC. Answer:14cm 130. From the top of a 200 meters high building, the angle of depression to the bottom of a second building is 20 degrees. From the same point, the angle of elevation to the top of the second building is 10 degrees. Calculate the height of the second building. Answer:296.89m 131. In the right triangle ABC, where AB=3m and BC=6m. Find the hypotenuse. Answer:6.71m 132. In the right triangle ABC, where the hypotenuse AC=2.5m and BC=5m. Find the AB Answer: 5.59m 133. A nursery plants a new tree and attaches a guy wire to help support the tree while its roots take hold. An eight foot wire is attached to the tree and to a stake in the ground. From the stake in the ground the angle of elevation of the connection with the tree is 42º. Find the height of the connection point on the tree. Answer:13ft 134. From the top of a fire tower, a forest ranger sees his partner on the ground at

an angle of depression of 40º. If the tower is 45 feet in height, how far is the partner from the base of the tower? Answer:53.6ft

cos(2x) - sin(2x) ] = cos(x) sin(3x) Answer: cos(x) - cos(x) sin(2x) sin(x) cos(2x)

135. Find the shadow cast by a 10 foot lamp post when the angle of elevation of the sun is 58º. Find the length to the nearest tenth of a foot. Answer:6.2ft

143. Find sin(x) and tan(x) if cos(pi/2 - x) = - 3/5 and sin(x + pi/2) = 4/5? Answer:-2 sin(x - pi/3) + 2

136. A ladder leans against a brick wall. The foot of the ladder is 6 feet from the wall. The ladder reaches a height of 15 feet on the wall. Find to the nearest degree, the angle the ladder makes with the wall. Answer:22degrees 137. A radio station tower was built in two sections. From a point 87 feet from the base of the tower, the angle of elevation of the top of the first section is 25º, and the angle of elevation of the top of the second section is 40º. To the nearest foot, what is the height of the top section of the tower? Answer: 32.43ft 138. Two friends, Angelo and Koie started climbing a pyramidshaped hill. Angelo climbs 315 mtr and finds that the angle of depression is 72.3 degrees from his starting point. How high he is from the ground Answer:300.20m 139. A man is observing a pole of height 55 foot. According to his measurement, pole cast a 23 feet long shadow. Can you help him to know the angle of elevation of the sun from the tip of shadow? Answer:67.3degrees 140. Linda measures the angle of elevation from a point on the ground to the top of the tree and find it to be 35°. She then walks 20 meters towards the tree and finds the angle of elevation from this new point to the top of the tree to be 45°. Find the height of the tree. (Round answer to three significant digits) Answer:46.7m 141. From the top of a cliff 200 meters high, the angles of depression of two fishing boats in the same line of sight on the water are 13 degrees and 15 degrees. How far apart are the boats? (Round your answer to 4 significant digits) Answer:119m 142. Prove that [ cos(x) - sin(x) ][

144. Find the exact value of [ tan (25°)+ tan (50° ] / [ 1 tan( 25°) tan(50°) ] Answer:3.73 145. What is the angle B of triangle ABC, given that A = 46°, b = 4 and c = 8?(Note: side a faces angle A, side b faces angle B and side c faces angle C). Answer: 29degrees 146. Find the exact value of tan (s + t) given that sin s = 1/4, with s in quadrant 2, and sin t = -1/2, with t in quadrant 4. Answer: -0.278 147. Find all angles of a triangle with sides 9, 12 and 15. Answer: 53degrees 147. Write an equation for a sine function with an amplitude of 5/3 , a period of pi/2, and a vertical shift of 4 units up. Answer: 5/3 sin(4x)+4 148. Find the exact values of cos (13π/12). Answer:1.17 150. Two gears are interconnected. The smaller gear has a radius of 4 inches, and the larger gear has a radius of 10 inches. The smaller gear rotates 890 degrees in 4 seconds. What is the angular speed, in degrees per minute, of the larger rotate? Answer:5340degrees/min

15'. Calculate the length of the sides AD Answer:4.5877m 154.The radius of a circle measures 25 m. Calculate the angle between the tangents to the circle, drawn at the ends of a chord with a length of 36 m. Answer:87.89degrees 155. Find tthe area of the triangle if the base is 21m and the height is 15m. Answer: 157.5sqr.m 156. Find the hypotenuse of the triangle ABC. If AB=10m and BC=15m Answer:18.03m 157. A hiker is hiking up a 12 degrees slope. If he hikes at a constant rate of 3 mph, how much altitude does he gain in 5 hours of hiking? Answer: 3.12m 158. A ramp is pulled out of the back of truck. There is a 38 degrees angle between the ramp and the pavement. If the distance from the end of the ramp to to the back of the truck is 10 feet. How long is the ramp? Answer:12.69ft 159. Neil sees a rocket at an angle of elevation of 11 degrees. If Neil is located at 5 miles from the rocket launch pad, how high is the rocket? Answer:97miles 160. A balloon is hovering 800 ft above a lake. The balloon is observed by the crew of a boat as they look upwards at an angle of 0f 20 degrees. 25 seconds later, the crew had to look at an angle of 65 degrees to see the balloon. How fast was the boat traveling? Answer: 49.75miles/hr 161. A telephone pole casts a shadow that is 18 ft. long. If the angle of elevation of the sun is 68°, what is the height of the pole in ft? Answer:44.55ft 162. Two buildings are 300 ft apart. If the angle of elevation from the top of the shorter building to the top of the taller building is 10°, what is the difference in the height of the two buildings? Answer:52.8ft

151. Solve for angle C of the oblique triangle ABC given, a = 80°, c = 115° and A = 72° Answer: 119.93degrees

163. From a point on the ground 96 m from a tree, the angle to the top of the tree is 38 degrees. What is the height of the tree? Answer:31.19m

152. Calculate the radius of the circle circumscribed in a triangle, where A = 45 °, B = 72 ° and a = 20m. Answer:14.14m

164. The angle form the ground to the top of the Statue of Liberty is 7 degrees at a distance of 1220 ft from the building. Find the height of the statue. Answer:149.80ft

153 The diagonals of parallelogram ABCD measure 10 cm (AC) and 12 cm (BD), and the angle that they form in the centre is 48°

165. A submarine maintains a diving angle of 22. How far has it travelled when it is directly under a point 350 m along the surface from the point where it submerged?

Answer:178.33m 166.  A surveyor wants to find the distance between peaks A and B. He finds point C, 288 ft from peak A, so that ACB is a right angle. The measure BAC is 89. Find the distance AB. Answer:273.90ft 167. Sears Tower is 1454 ft tall. Suppose point A is 1000 ft from the base of the tower. What is the tangent of the angle at A formed by the ground and the line of vision to the top of the tower? Answer:55.48degrees 168. Diving at a constant angle A, a submarine descends 102 m while travelling 300 m. Find the degree measure of A. Answer:18.78degrees 169. The triangle ABC is a right triangle. AB=2m and BC=7m. Find the hypotenuse. Answer: 7.28m 170. The triangle ABC is a right triangle. BC=5meters and angle A=7degrees. Find AB. Answer:40.72 171. Find the height of the right triangle if the area is 300msqrd and the base is 21m. Answer: 28.57m 172. Find the area of the triangle if the base is 30m and the height is 35m. Answer: 525sqr.m Situation: A ladder of length 20 meters is resting against the wall. The base of the ladder is x meters away from the base of the wall and the angle made by the wall and the ladder is t. 173. Find x in terms of t. Answer: 74.64m 174. Starting from t = 0 (the ladder against the wall) and then gradually increase angle t; for what size of angle t will x be the quarter of the length of the ladder? Answer:14degrees Situation: A Ferris wheel with a radius of 25 meters makes one rotation every 36 seconds. At the bottom of the ride, the passenger is 1 meter above the ground. 175. Let h be the height, above ground, of a passenger. Determine h as a function of time if h = 51 meter at t = 0. Answer:26m

176. Find the height h after 45 seconds. Answer: 26m Situation: A ladder is leaning up against a house. The bottom of the ladder is 3 ft away from the building and the ladder makes an angle of 75 degrees with the ground. 177. How high up the building does the ladder reach? Answer:11.20m 178. How long is the ladder? Answer:11.59m Situation: 179. After takeoff, an airplane maintained a flight angle of 8 degrees with the ground. Find the elevation after it covered after it covered a ground distance of 1200 m. Answer:168.65m 180 Find the distance it traveled in the air along the flight path while covering the ground distance of 1200 m. Answer:1211.75m 181. Find the supplement of an angle whose compliment is 62° Answer: 152° 182. The central angle has a supplement five times its compliment. Find the angle. Answer: 67.5° 183. The sum of the two interior angles of the triangle is equal to the third angle the difference of the two angles is equal to 2/3 of the third angle. Find the third angle. Answer: 90° 184. The measure of 1½ revolutions counterclockwise is: Answer: 540° 185. The measure of 2.25 revolutions counterclockwise is: Answer: 810° 186. Solve for θ: sin θ – sec θ + csc θ – tan 2θ = -0.0866. Answer: 46° 187. What are the exact values of the cosine and tangent trigonometric functions of acute angle A, given that sin A = 3/7.

Answer:

3 √ 10 20

188. Given three angles A, B and C whose sum is 180°. If tan A + tan B + tan C = x, find the value of tan A x tan B x tan C. Answer: x 189. What is the sine of 820°? Answer: 0.984 190. Csc 270°? Answer: -1 191. If conversine θ is 0.134, find the value of θ. Answer: 60° 192. Solve for cos 72° if the given relationship is cos 2A = 2 cos2 A – 1. Answer: 0.309 193. If sin 3A = cos 6B then: Answer: 30° 194. Find the value of sin (arccos 15/17). Answer: 8/17 195. Find the value of cos [arcsin (1/3) + arctan (2/√5)]. Answer:

2 ( √10−1) 9

196. If sin 40° + sin 20° = sin θ, find the value of θ. Answer: 80° 197. How many different value of x from 0° to 180° for the equation (2 sin x – 1) (cos x + 1) = 0? Answer: 3 198. For what value of θ (less than 2π) will the following equation be satisfied? sin2 θ + 4 sin θ + 3 = 0. Answer: 3π/2 199. Find the value of x in the equation csc x + cot x = 3. Answer: π/5 200. If sec2 A is 5/2, the quantity 1 – sin2 A is equivalent to: Answer: 0.4 201. Find sin x if 2 sin x + 3 cos x – 2 = 0. Answer: 1 and -5/13 202. If sin A = 4/5, A is quadrant II, sin B = 7/25, B in quadrant I, find sin (A + B). Answer: 3/5 203. If sin A = 2.571x, cos A = 3.06x, and sin 2A = 3.939x, find the value of x. Answer: 0.250 204. If cos θ = √3 / 2, then find the value of x if x = 1 – tan2 θ. Answer: 2/3 205. If sin θ – cos θ = -1 / 3, what is the value of sin 2θ?

Answer: 8/9 206. If x cos θ + y sin θ = 1 and x sin θ – y cos θ = 3, what is the relationship between x and y? Answer: x2 + y2 = 10 207. If sin x + 1 / sin x = √2, then sin2 x + 1 / sin2 x is equal to: Answer: 0 208. The equation 2 sin θ + 2 cos θ – 1 = √3 is: Answer: A conditional equation 209. If x + y = 90°, then

sin x tan y is equal to: sin y tan x

Answer: cot x 210. If cos θ = x / 2 then 1 – tan2 θ is equal to: Answer: (2x2-4)/x2 211. Find the value in degrees of arccos (tan 24°). Answer: 63.56° 212. Arctan [2 cos (arcsin (√3 / 2))] is equal to: Answer: π/4 213. Solve for x in the equation: arctan (2x) + arctan (x) = π/4. Answer: 0.281 214. Solve for x from the given trigonometric equation: arctan (1-x) + arctan (1+x) = arctan 1/8. Answer:

+¿ ¿ −¿ 4 ¿

215. Solve for y if y = (1/sinx – 1 /tan x) (1 + cos x). Answer: sin x 216. Solve for x: x = (tan θ + cot θ)2 sin2 θ – tan2 θ. Answer: 1 217.Solve for x: x = 1 – (sin θ – cos θ)2. Answer: sin 2θ 218. Simplify cos4 θ – sin4 θ. Answer: 2 cos2 θ-1 219. Solve for x: x=

1−tan 2 a 1+tan 2 a

Answer: cos 2a 220. cos3 x – sin3 x =? Answer: (cos x – sin x) (cos2 x + cos x sin x + sin2 x) 221. Find the value of y: y = (1 + cos 2θ) tan θ. Answer: sin 2θ 222. The equation 2sinh x cosh x is equal to: Answer: sinh 2x

223. Simplifying the equation sin2 θ (1 + cot2 θ) gives: Answer: 1

side c if angle C = 100°, side b = 20, and side a = 15. Answer: 27

224. Find the value of sin (90° + A). Answer: cos A

237. In triangle ABC, A = 45° and angle C = 70°. The side opposite angle C is 40 m long. What is the side opposite angle A? Answer: 30.1 m

225. Which of the following expression is equivalent to sin 2θ? Answer: 2 sin θ cos θ 226. If tan θ = x2, what is the value of sin θ? Answer: x2/ 1+ x 4

√

227. In isosceles right triangle, the hypotenuse is how much longer than its sides? Answer: √2 228. Find the angle mils subtended by a line 10 yards long at a distance of 5000 yards. Answer: 2.04 mil 229. The angle or inclination of ascend of a road having 8.25% grade is ___ degrees. Answer: 4.716° 230. The sides of a right triangle is in arithmetic progression whose common difference is 6 cm. Its area is: Answer: 216 cm2 231. The hypotenuse of a right angle is 34 cm. Find the length of the shortest leg if it is 14 cm shorter than the other leg. Answer: 16 cm 232. A truck travels from point M northwards 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? Answer: 47.88 km 233. Two sides of a triangle measures 6 cm. and 8 cm. and their included angle is 40°. Find the third side. Answer: 5.144 cm 234. Given a triangle: C = 100°, a = 15, b = 20. Find c: Answer: 27 235. Given angle A = 32°, angle B = 70°, and side c = 27 units. Solve for side a of the triangle. Answer: 14.63 units 236. In a triangle, find the

238. Two sides of a triangle are 50 m and 60 m long. The angle included between these sides is 30°. What is the interior angle (in degrees) opposite the longest side? Answer: 93.74° 239. 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. Answer: 4.73 cm 240. If AB = 15 m, BC = 18 m, and CA = 24 m, find the point intersection of the angular bisector from the vertex C. Answer: 14.3 cm 241. In triangle ABC, angle C = 70°, angle A = 45°, AB = 40 m. What is the length of the median drawn from vertex A to side BC? Answer: 36.28 m 242. 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: Answer: 52.23 units 243. Given triangle ABC whose angles are A = 40°, B = 95° and side b = 30 cm. Find the length of the bisector of angle C. Answer: 21.74 cm 244. 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: Answer: 125 m 245. From a point outside of an equilateral triangle, the distance to the vertices are 10 m, 10 m, and 18 m. Find the dimension of the triangle. Answer: 19.94 m 246. 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° N of W and from B the bearing of C is 26° N of E. Approximate the shortest distance of tower C to the highway. Answer: 273.92 m 247. An airplane leaves an aircraft carries 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? Answer: 2.13 hrs

248. 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? Answer: 2√5 m 249. From the top of the building 100 m, high the angle of the depression of a point A due to East of it is 30°. From a point B due to south of the building, the angle of elevation of the top is 60°. Find the distance AB. Answer: 100√3/3 250. An observer found the angle of elevation of the top of the tree to be 27°. After moving 10 m closer (on the same vertical and horizontal plane as the three), the angle of elevation becomes 54°. Find the height of the tree. Answer: 8.09 m 251. 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 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: Answer: 12,493 feet 252. A 50-meter vertical tower casts a 62.3-meter shadow when the angle of elevation of the sun is 41.6°. The inclination of the ground is: Answer: 4.72° 253. 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. Answer: 11 m 254. A pole cast a shadow of 15 meters long when the angle of elevation of the sun 61°. If the pole has leaned 15° from the vertical directly toward the sun, what is the length of the pole? Answer: 54.23 m 255. 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 ft apart, at the same elevation on a direct line with the tower. The vertical angle at point A is 30° and at point B is 40°. What is the height of the tower? Answer: 92.54 ft

256. From the top of tower A, 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 high is tower A? Answer: 40.71 m 257. 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? Answer: 259.3 m 258. 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? Answer: 73.61 m 259. The angle of elevation of a point C is 29°42’: the angle of elevation of C from the another point A 31.2 m directly below B is 59°23’. How high is C from the horizontal line through A? Answer: 47.1 m 260. A rectangular 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: Answer: P601,650.00 261. A man improvises a temporary shield from the sun using triangular piece of wood with dimensions of 1.4 m, 1.5 m, 1.3 m, with the longer side lying horizontally on the ground, he props up the 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? Answer: 0.5 m2 262. A rectangular piece of wood 4 cm x 12 cm tall is tilted at an angle of 45°. Find the vertical distance between the lower corner and upper corner. Answer: 8√2 263. 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. Answer: 9.17 inches 264. If the bearing of A from B is S 40° W, then the bearing of B from A is: Answer: N 40° E 265. A plane hillside is inclined at an angle of 28° with the horizontal. A man wearing skis can climb this hillside by following a straight path inclined at an angle of 12° to the horizontal, but one without skis must follow a path inclined at an angle of only 5° with the horizontal. Find the angle between the directions of the two paths. Answer: 15.56° 266. Calculate the area of a spherical triangle whose radius is 5 m and whose angles are 40°, 65°, and 110°. Answer: 15.27 sq. m 267. A right spherical triangle has an angle C = 90° a = 50°, and c = 80°. Find the side b. Answer: 74.33° 268. If the time 8:00 A.M. GMT, what is the time in the Philippines, which is located at 120° East longitude? Answer: 4 p.m. 269. An airplane flew from Manila (14°36’N, 121°05’E) at a course of S 30° E maintaining a certain altitude and following a great circle path. If its ground speed is 350 knots, after how many hours will it cross the equator? Answer: 2.87 hrs 270. Find the distance in nautical miles between Manila and San Francisco. Manila is located 14° 36’ N latitude and 121°05’ E longitude. San Francisco is situated 37°48’ N latitude and 122° 24’ W longitude. Answer: 6046.2 nautical miles 271. A at certain point on the ground, the tower at the top of 20-m high building subtends an angle of 45°. At another point on the ground 25 m closer the building, the tower subtends an angle of 45°. Find the height of the tower. Answer: 101.85 m 272. A wooden flagpole is embedded 3 m deep at corner A of a concrete horizontal slab ABCD, square in form and measuring 20 ft on a side. A storm broke the flag pole at a point one meter above the slab and inclined toward corner C in the direction of the diagonal AC. The vertical angles observed at the center of the slab and at corner C to the tip of the flag pole were 65° and 35°, respectively. What is the total length of the flagpole above the slab in yards? Answer: 5.61 yards 273. From the third floor window of a

building, the angle of depression of an object on the ground is 35°58’, while from a sixth floor window, 9.75 m above the first point of observation the angle of depression is 58°35’. How far is the object from the building? Answer: 10. 7 m 274. The sides of a triangle are 18 cm, 24 cm, and 34 cm, respectively. Find the length of the median to the 24 cm side, in cm. Answer: 24.41 cm 275. In the spherical triangle ABC, A = 116°19’, B = 55°30’, and C = 80°37’. What is the value of side a. Answer: 115.573° 276. Find the supplement of an angle whose compliment is 42° Answer: 138° 277. The central angle has a supplement three times its compliment. Find the angle. Answer: 45° 278. The measure of 2½ revolutions in degrees is: Answer: 900°

Answer: 2 cos2 x – 1 289. Simplify 1 – 2 sin2 –x. Answer: 2 cos2 x – 1 290. If tan θ = x2, find cos θ. Answer: 1/ 1+ x 4

√

291. In problem 290, Find sec θ. Answer: 1+ x 4

√

292. In problem 290, Find csc θ. Answer: 1+ x 4 /x2

√

293. ½ revolutions is equal to what degree? Answer: 180° 295. 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 30 ft apart, at the same elevation on a direct line with the tower. The vertical angle at point A is 30° and at point B is 40°. What is the height of the tower? Answer: 55.52 ft

279. In triangle ABC, angle A = 30°, and angle B = 85°. Find angle C. Answer: 65°

296. In triangle XYZ X= 55°, Y= 20°, find Z. Answer: 105°

280. If cos x = 0.087. Find sin x. Answer: 85°

297. Find cos θ if its supplementary angle is 100° Answer: 0.1736

281. What is 45° in radian?

1 Answer: π 4

298. Find cot θ if tan θ = -0.8391 Answer: -1.192

282. What is sec 180°=? Answer: -1

θ.

283. Given angle A = 40°, angle B = 80°, and side c = 19 units. Solve for side b of the triangle. Answer: 21.61 units 284. In problem 283, solve for side a of the triangle. Answer: 14.10 units 285. A triangle AOB is a right triangle. Line OB is the hypotenuse, AB as the longer side and OA as the shorter side. If the line AB is 18 m and line OA is 7 m. Line OB is: Answer: 19.3 m 286. sin x = 0.9063, find csc x. Answer: 1.1034 287. Find the angle if its tan is equal to 0.17633. Answer: 190° 288. Simplify cos4 x – sin4 x.

299. sin θ = 0.5, find tangent Answer: √3/3 300. What is 2400 mils in degrees? Answer: 135°