TRIGONOMETRY
I
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
ENGINEERING MATHEMATICS
C2 - 1
I
TRIGONOMETRY
I
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
ENGINEERING MATHEMATICS
C2 - 2
I
TRIGONOMETRY
I
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
ENGINEERING MATHEMATICS
C2 - 3
I
TRIGONOMETRY
I
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
ENGINEERING MATHEMATICS
C2 - 4
I
TRIGONOMETRY
I
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
______.
ENGINEERING MATHEMATICS
C2 - 5
I
TRIGONOMETRY
I
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
ENGINEERING MATHEMATICS
C2 - 6
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TRIGONOMETRY
I
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
ENGINEERING MATHEMATICS
C2 - 7
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TRIGONOMETRY
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
ENGINEERING MATHEMATICS
C2 - 8
I
TRIGONOMETRY
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
ENGINEERING MATHEMATICS
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TRIGONOMETRY
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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
ENGINEERING MATHEMATICS
C2 - 10
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TRIGONOMETRY
I
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
ENGINEERING MATHEMATICS
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I
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.
ENGINEERING MATHEMATICS
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TRIGONOMETRY
I
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.
ENGINEERING MATHEMATICS
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TRIGONOMETRY
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:
ENGINEERING MATHEMATICS
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I
TRIGONOMETRY
I
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:
ENGINEERING MATHEMATICS
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I
TRIGONOMETRY
I
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:
ENGINEERING MATHEMATICS
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I
TRIGONOMETRY
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
ENGINEERING MATHEMATICS
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I
TRIGONOMETRY
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:
ENGINEERING MATHEMATICS
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I
TRIGONOMETRY
I
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
ENGINEERING MATHEMATICS
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I
TRIGONOMETRY
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:
ENGINEERING MATHEMATICS
C2 - 20
I
TRIGONOMETRY
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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