CERTC - EE REFRESHER 1 MATH

Instruction: Select the correct answer for each of the following 26. A man drives 500 m along a road which is inclined 20 deg to the
questions. Mark only one answer for each item by shading the box horizontal. How high above his starting point is he? (171 m)
corresponding to the letter of your choice on the answer sheet
27. A tree broken over by the wind forms a right triangle with the
provided.
ground. If the broken part makes an angle of 50 deg with the
ground and the top of the tree is now 20 ft from its base, how tall
Strictly no erasures allowed. Use pencil no. 1 only.
was the tree? (55ft)
NOTE: Whenever you come across a caret (^) sign, it means
exponentiation. 28. Two straight roads intersect to form an angle of 75 deg. Find the
shortest distance from one road to a gas station on the other
BASIC PROPERTIES road 1000m from the junction. (3732 m)
1. The minute hand of a clock is 12 cm long. How far does the tip of 29. Two buildings with flat roofs are 60m apart. From the roof of the
the hand move during 20 min? (25.1 cm) shorter building, 40 m in height, the angle of elevation to the
2. A central angle of a circle of radius 30 cm intercepts an arc of 6 edge of the roof of the taller building is 40 deg. How high is the
cm. What is the central angle in radian? (0.2 rad) taller building? (90 m)
3. A railroad curve is to be laid out on a circle. What radius should be 30. A ladder with its foot in the street makes an angle of 30 deg with
used if the track is to change direction by 25 deg in a distance of the street when its top rests on a building on one side of the
120m? (275 m) street and makes an angle of 40 deg when its top rests on a
building on the other side of the street. If the ladder is 50 ft long,
4. A train moving at a rate of 8mph along a piece of circular track of
how wide is the street? (82 ft)
radius 2500 ft. Through what angle does it turn in one minute?
(16.13 deg) 31. Find the perimeter of an isosceles triangle if its base is 40 cm
and the base angle is 70 deg. (157 cm)
5. A sector of a circle has a central angle of 50 deg and an area of
605 sq. cm. Find the radius of the circle. (37.2 cm) 32. Considering the earth as a sphere of radius 3960 miles, find the
radius of the 40th parallel of latitude. (3033 ft)
6. The end of a 40-in pendulum describes an arc of 5 in. Through
what angle does the pendulum swing? (7.16 deg) 33. Find the perimeter of a regular octagon inscribed in a circle of
radius 150 cm. (918 cm)
7. A reversed curve of a railroad track consists of two circular arcs.
The central angle of one is 20 deg with radius 2500 ft and the 34. To find the width of a river, a surveyor set up his transit at a point
C on one bank and sighted across a point B on the opposite
central angle of the other is 25 deg with rdius 3000ft. Find the total
length of the two arcs. (2182 ft) bank; then turning an angle 90 deg, he laid off a distance of
225m on point A. Finally, setting the transit at A, he measured
8. If an automobile wheel 78 cm in diameter rotates at 600 rpm,
that angle CAB is 48 deg 20 min. Find the width of the river. (253
what is the speed if the car in kph? (88.2 kph)
m)
TRIGO FUNCTIONS
35. Points A and D are on both sides of a swamp. A surveyor wants
9. Find the first coterminal angle of 125 deg. (485 deg)
to locate a point B along the line on the opposite side of the
10. Find the first negative coterminal angle of pi/6 rad. (-11pi/6) swamp. To do this, at A, he turned at an angle of 51 deg 16 min
towards point C 1585 ft away. Then, at point C, he turned 90 deg
11. Determine the angle subtended by the point (-3,4) from the
from line AC towards point B. How far is C from B? (1976 ft)
positive x-axis. (126.87 deg)
36. From a point A on level ground, the angles of elevation of the top
12. Determine the angle subtended by the point (-3,4) from the x-
D and the bottom B of a flagpole situated on top of a hill are
axis. (53.13 deg)
measured as 47 deg 54 min and 39 deg 45 min, respectively.
13. Find the value of cosθ if sinθ=8/17 and θ is in Quadrant 1. Find the height of the hill if the height of the flagpole is 115.5 ft.
(15/17) (350 ft.)
14. Find the value of sin θ if cosθ=5/6 and θ is in Quadrant IV. (- 37. From the top of a lighthouse, 175 ft above the water, the angle of
sqrt(11)/6) depression of a boat due south is 18 deg 50 min. Calculate the
15. Find sin θ, given that cosθ= - 4/5 and tanθ is positive. (-3/5) speed of the boat if, after it moves due west for 2 min, the angle
of depression is 14 deg 20 min. (227 ft/min)
16. In what quadrant will θ terminate if its tangent is positive and its
secant negative? (3) 38. A wheel, 5 ft in diameter, rolls up an incline of 18 deg 20 min.
What is the height of the center of the wheel above the base of
17. At what point on the unit circle will the angle 17pi terminate? (- the incline if the wheel has rolled 5 ft up the incline? (3.95 ft)
1,0)
39. A wall is 15 ft high and 10 ft from a house. Find the length of the
18. At what quadrant will θ terminates if cos θ and cotθ are both shortest ladder which will just touch the top of the wall and reach
negative? (2) a window 20.5 ft above the ground. (42.5 ft)
RIGHT TRIANGLES 40. A parked car is spotted from a hotel window which is 100 m
19. A support wire is anchored 12 m up from the base of a flagpole above the car. If the angle of depression from the window to the
and the wire makes 15 deg with the ground. How long is the car is 15.4 deg, how far is the car from the window? (377 m)
wire? (46m) 41. A tower 45 m high casts a shadow of 12m. what is the angle of
20. When the sun is 20 deg above the horizon, how long is the elevation of the sun? (75 deg)
shadow cast be a building 50 m high? (137m) 42. A rectangle is 45 cm long and 34 cm wide. What is the angle the
21. From the top of a lighthouse 120 m above sea level, the angle of diagonal makes with the longer side? (35 deg)
depression of a boat is 15 deg. How far is the boat from the 43. A railroad track makes an angle of 4.8 deg with the horizontal.
lighthouse? (448m) How far must the train go up the track for it to gain 30 m in
22. Find the length of the chord of a circle of radius 20cm subtended altitude? (359 m)
by a central angle of 150 deg. (39cm)
PRACTICAL APPLICATIONS
23. Find the height of a tree if the angle of elevation of its top 44. A motorboat moves in a direction N40°E for 3h at 20mph. how far
changes from 20 deg to 40 deg as the observer advances 75 ft north does it travel? (46 miles)
towards the tree. (48 ft)
45. Three ships are situated as follows: A is 225 miles due North of
24. A tower on level ground is due north of point A and due west of C, and B is 375 miles due east of C. What is the bearing of B
point B, a distance of 3000 ft. The angle of elevation of the tower from A? (S59°E)
from A is 30 deg while at B it is 20 deg. Find the height of the
tower. (284 ft) 46. Three ships are situated as follows: A is 225 miles west of C
while B, due south of C, bears S25°10’E from A. How far is B
25. Four holes are bored on a circular plate. If the radius of the plate from A? (529 miles)
id 20 in and the holes are well-spaced, find the distance between
centers of the holes. (28.28 in)

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 CERTC - EE REFRESHER 1 MATH

47. From a boat sailing due north at 16.5 kph, a wrecked ship and an 67. A 150-lb shell is dragged up a runway inclined 40 deg to the
observation tower are observed in a line due east. One hour horizontal. Find the force of the shell against the runway and the
later, the wrecked ship and the tower have bearings S34°40’E force required to drag the shell. (115 lb, 96lb)
and S65°10’E. Find the distance between the wrecked ship and
68. A steel ball, weighing 50 kg, is on an inclined surface. What
the tower. (24.2 km)
angle does the inclined surface make with thehorizontal if a force
48. A ship sailing due east when a light is observed bearing of 10 kg is exerted on the inclined surface? (12 deg)
N62°10’E. After the ship has travelled 2250m, the light bears
N48°25’E. if the course is continued, what is the closest distance TRIGO IDENTITIES
at which the ship approaches the light? (2934m) cos 𝜃
69. Simplify 𝑡𝑎𝑛𝜃 + . (secθ)
1+sin 𝜃
49. An object is acted upon by two forces, one 150 lb due north and 70. Simplify (𝑠𝑖𝑛𝜃 + 𝑐𝑜𝑠𝜃)2 + (𝑠𝑖𝑛𝜃 − 𝑐𝑜𝑠𝜃)2 . (2)
the other 200lb due east. Find the magnitude and direction of the 71. Simplify (sin 𝜃)2 (1 + (cot 𝜃)2 ) (1)
resultant. (250 lb, N53°10’E) 72. Simplify ( )
50. An airplane is moving horizontally at 240mph when a bullet is
shot with speed 2750 fps at right angle to the path of the OBLIQUE TRIANGLES
airplane. Find the angle that the path of the bullet makes with the 73. In triangle ABC, given c = 25, A = 35 deg, and B = 68 deg, find a.
path of the airplane. (82.7 deg) (15)
51. A river flows due south at 125 ft/min. A motorboat, moving at 475 74. Solve for c in triangle ABC, given a = 31.5, b = 51.8, and A =
ft/min in still water, is headed due east across the river. Find the 33°40'. (56.1 or 30.1)
direction in which the boat moves and its speed. (491 ft/min, 75. Solve for c in triangle ABC, given a = 132, b = 224, and C =
S75°20’E) 28°40'. (125)
52. In the previous question, whatdirection must the boat be headed 76. Solve for b in triangle ABC, given a = 322, c = 212, and B =
in order that it move due east and what is its speed in that 110°50'. (444)
direction?(N74°40’E, 458 ft/min)
53. A telegraph pole is kept vertical by a guy wire which makes an 77. A and B are two points on opposite banks of a river. From A, a
angle of 25 deg with the pole and whichexerts a pull of F = 300 Ib line AC = 275 m is laid off and the angles CAB = 125°40' and
on the top. Find the horizontal component of the pull. (127 lb) ACB = 48°50' are measured. Find the length of AB. (2160m)

54. A guy wire 78 ft long runs from the top of a telephone pole 56 ft 78. A tower 125 ft high is on a cliff on the bank of a river. From the
high to the ground and pulls on the pole witha force of 290 Ib. top of the tower the angle ofdepression of a point on the opposite
What is the horizontal pull on the top of the pole? (201 lb) shore is 28°40' and from the base of the tower the angle
ofdepression of the same point is 18°20'. Find the width of the
55. A man pulls a rope attached to a sled with a force of 100 lb. The river and the height of the cliff. (580ft, 192 ft)
rope makes an angle of 27 deg with the ground. Find the
effective pull tending to move the sled along the ground? (89 lb) 79. A pilot wishes a course 15°0' against a wind of 25 mph from
160°30'. Find his required heading andthe groundspeed when
56. A plane is headed due east with airspeed 240 km/h. If a wind at the airspeed is 175 mph. (195 mph, 140°50’)
40 km/h from the north is blowing, find thegroundspeed and
course. (243kph, S80°30’E or 99deg 30min) 80. Two forces of 17.5 and 22.5 lb act on a body. If their directions
make an angle of 50°10' with eachother, find the magnitude of
57. An airplane is flying on a heading of 45 deg with an airspeed of their resultant and the angle that it makes with the larger force.
500 mph and the wind is blowing on a heading of 315 deg at a (36.3 lb, 21°40’)
speed of 75 mph. Find the drift angle, the groundspeed, and the
course of the plane. (8.5 deg, 506mph, 36.5 deg) 81. From A a pilot flies 125 km in the direction N38°20'W and turns
back. Through an error, the pilotthen flies 125 km in the direction
58. The heading of an airplane is 75 deg and the airspeed is 200 S51°40'E. How far and in what direction must the pilot now fly to
mph. Find the groundspeed and course ifthere is a wind of 40 reach the intended destination A? (S45°20’W, 29km)
mi/h from 165 deg. (204 mph, 63°40’)
82. The distances of a point C from two points A and B, which cannot
59. The airspeed of an airplane is 200 km/h. There is a wind of 30 be measured directly, are required. The line CA is continued
km/h from 270 deg. Find the heading andgroundspeed in order to through A for a distance 175 m to D, the line CB is continued
track 0 deg. (198kph, 351°20’) through Bfor 225 m to E, and the distances AB = 300 m, DB =
60. There is a wind of 35 mi/h from 320 deg. Find the airspeed and 326 m, and DE = 488 m are measured. Find AC. (145 m)
heading in order that the groundspeed and course be 250 mi/h 83. Two ships have radio equipment with a range of 200 km. One is
and 50 deg, respectively. (252 mph, 42 degs) 155 km N42°40'E and the other is 165 km N45°10'W of a shore
61. An aviator heads his airplane due west. He finds that because of station. Can the two ships communicate directly? How far are
a wind from the south, the course makes an angle of 20 deg with they? (No, 222km)
the heading. If his airspeed is 100 mi/h, what is his groundspeed 84. A ship sails 15.0 mi on a course S40'1O'W and then 21.0 mi on a
and what is the speed of the wind? (106 mph, 36 mph) course N28'20'W. Find the distance anddirection of the last
62. An airplane is headed west while a 40 mi/h wind is blowing from position from the first. (209mi, N70°30’W)
the south. What is the necessary airspeed to follow a course 85. A lighthouse is 10 km northwest of a dock. A ship leaves the
N72°W and what is the groundspeed? (123 mph, 129 mph) dock at 9 A.M. and steams west at 12 kmh. Atwhat time will it be
63. A barge is being towed north at the rate 18 mi/h. A man walks 8 km from the lighthouse? (9:17AM and 9:54AM)
across the deck from west to east at the rate 6ft/s. Find the 86. Two forces of 115 and 215 Ib acting on an object have a
magnitude and direction of his actual velocity. (27fps, N12°50’E) resultant of magnitude 275 Ib. Find the angle betweenthe
64. A ship at A is to sail to C, 56 km north and 258 km east of A. directions in which the given forces act. (70 deg 50 min)
After sailing N25°10'E for 120 mi to P, the shipis headed toward 87. A tower 150 m high is situated at the top of a hill. At a point 650
C. Find the distance of P from C and the required course to reach m down the hill the angle between the surfaceof the hill and the
C. (214km, S75°40’E) line of sight to the top of the tower is 12°30'. Find the inclination
of the hill to a horizontal plane. (7 deg 50 min)
65. A weight of 200 Ib is placed on a smooth plane inclined at an
angle of 37.6 deg with the horizontal and held inplace by a rope 88. Three circles of radii 115, 150, and 225 m, respectively, are
parallel to the surface and fastened to a peg in the plane. Find tangent to each other externally. Find the angles ofthe triangle
the pull on the string. (122 lb) formed by joining the centers of the circles. (43 deg 10min, 61
deg 20 min, 75 deg, 30 min)
66. A man wishes to raise a 300-lb weight to the top of a wall 20 m
high by dragging it up an incline. What is thelength of the shortest
inclined plane he can use if his pulling strength is 140 Ib? (43m)

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 CERTC - EE REFRESHER 1 MATH

TRIGO EQUATIONS
89. Solve for x in 2sinx-1=0 if x<pi. (pi/6 and 5pi/6)
90. Solve for x: sin^2 x +sinx -2 =0. (pi/2)
91. Solve for x: 3cos^2 x = sin^2 x ; x<pi/2. (pi/3)
92. Solve for x: cscx +cotx =sqrt(3). (pi/3)
93. Solve for x: 2cos^2 (x/2) = cos^2 x. (2.24, 4.04)

SPHERICAL TRIGONOMETRY (Additional Topic)

94. Area. What is the area of a spherical triangle ABC whose parts
are A=70 deg, B=80 deg, C=92 deg if the radius of the sphere is
64km. (4432 sq. km)
95. Right. Tan-Ad. Two sides of a spherical triangle are 86°40’ and
32°41’. Find the angle opposite of the given side.
96. Right. Co-Op. A spherical triangle ABC has the following given
parts: a=72°27’, b=61°49’ and C=90°. Find c.
97. Right. In a right spherical triangle, a=32 deg 16 min and c=119
deg 23 min. Solve for b. (125 deg 28 min)
98. Right. Given right spherical triangle ABC with C=90°, a=75°40’
and b=61°30’. Find A. (77.34 deg)
99. Right. Given spherical triangle with angles A=63°15’ and
B=135°34’. Find side b. (143.1 deg)
100. Right. An isosceles spherical triangle has the following parts
A=B= 80 deg, a=b=89 deg. Find c.(168.52 deg)
101. Latitude. Two cities are 270 miles apart lie on the same
meridian. What is the difference in latitude if the radius of the
earth is 3960 miles. (3.91 deg)
102. Latitude.If the earth is 3960 miles in radius, find the distance of
a point 36 deg N latitude from the equator. (2488 miles)
103. Nautical Miles. Find the distance in nautical miles between
A(40°30’N, 60°E) andB(80°20’N, 60°E). (7250 NM)
104. Nautical Miles. How far in statute miles is a place at latitude
40°N form the equator. (2764 mi)
105. Nautical Miles. How far apart are two points on the equator one
in longitude 40°E and the other in ongitude 150°W? (10,200NM)

106. Right Applications. An airplane flew from Manila (14°36’N,

121°5’E) on a course S32°E. At what longitude will it cross the
equator? (130°2’ E)
107. Right Applications. A ship leaves San Francisco (37°48’N,
122°26’W) and sailing along a great circle track, crosses the
equator at a point M whose longitude is 150°W. Find the distance
from M to San Francisco. (2732NM)
108. Time Difference. New York has a longitude of 74°W and
Portland has longitude of 122°W. Calculate their time difference.
(3.2 hours)
109. Oblique. Cosine Law for Sides. If an airplane is to fly from
Chicago (41°50’N, 87°37’W) to New York (40°43’N, 74°W) at an
average speed of 200NM/hr, approximately how long should the
trip take? (3hrs, 5 min)
110. Oblique. Cosine Law for Angles. Given spherical triangle ABC
with A=100 deg, B=70 deg and c=110 deg, find C. (255.1 deg)
111. Oblique. A ship sails from a place (20°N, 150°W). after 6 days,
it crosses the equator at longitude 170°W. find the speed of the
ship in NM/hr. (11.66)

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