TRIGONOMETRY
I
1. ECE Board April 1995 A pole cast a shadow 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 . 2!.1 ". 54.2! #. !4.!
. 21.4 degree ". 1!. degree #. 18. degree 6. ECE Board November 2002 A certain angle has has an e%plement e%plement 5 times the supplement/ ind the angle. A. 60.5 degree degree . 1,8 degree ". 1!5 degree #. 58.5 degree
2. ECE Board March 1996 $olve for % in the e&uation' arctan π
(2%) * arctan (%) +
4
0. ECE Board November 2002 ind the height of the tree if the angle of elevation of its top changes from 2, degrees to 4, degrees as the o3server advances 2! meters toward the 3ase. A. 1!.08 m . 16.08 m ". 14.08 m #. 15.08m
.
A. ,.2841 . ,.185. ". ,.218 #. ,.821 !. ECE Board March 1996 -he hypotenuse of a right triangle is !4 cm. ind the lengths of the two legs if one leg is 14 cm. longer than the other. A. 16 cm/ !, cm . 1! cm/ 20 cm ". 15 cm/ 2 cm #. 1, cm/ 14 cm
8. ECE Board November 2002 A wheel/ ! ft. in diameter/ diameter/ rolls down an inclined plane !, degrees with the horiontal. How high is the center of the wheel when it is 5 ft ft from the 3ase of the plane? A. 4 ft . 2.5 ft ". ! ft #. 5 ft
4. ECE Board March 1996 4
f sin A +
5
/ A in &uadrant / sin +
0 25
/ in &uadrant / find sin (A*) 0
A.
. ECE Board November 2002 f the complement of the angle theta is 25 of its supplement/ then theta is . . A. 45 degree degree . 05 degree ". 6, degree #. !, degree
5 !
.
5 2
".
5 !
#.
4
5. ECE Board March 1996 f 00 degree * ,.4, % + arctan (cot ,.25%)/ solve for %.
A. 2, degree degree
1,. ECE Board November 2002
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ENGINEERING MATHEMATICS I
TRIGONOMETRY
I
7ne side of the right triangle is 15 cm long and the hypotenuse is 1, cm longer than the other side. hat is the length of the hypotenuse? A. 1!.5 cm . 6.5 cm ". 12.5 cm #. 16.25 cm
". 15 #. ; 16. ECE board April 1999 $in ( < A) is e&ual to / when +20, degrees and A is an acute angle. A.
11. ECE BOard November 2003 hat is the 3ase of the logarithmic function log 4 + 2!? A. 8 . 2 ". ! #. 4
10. ECE Board April 1999 if sec2A is 52/ the &uantity 1=
sin2A is e&uivalent to A. ,.4 . ,.8 ". ,.6 #. 1.5
12 ECE Boars November 2003 A transmitter with with a height of 15 m is located on top of a mountain/ which is !., 9m high. hat is the farthest distance on the surface of the earth that can 3e seen from the top of the mountain? -a:e the radius of the earth to 3e 64,, 9m. A. 2,5 9m. . 225 :m. ". 152 :m. #. 2,,.82 :m.
18. ECE Board April 1999
cosA
4
- sinA
4
is e&ual to .
A. "os4A . $in 2A ". "os 2A #. $in 4A 1. ECE Board April 1999 7f what &uadrant is A/ if sec A is positive and csc A is negative? A. > . ". #.
1!. ECE Board November 2003 f y + arcsec (negative s&uare root root of 2)/ what is the value of y in degree? A. 05 . 6, ". 45 #. 1!5
2,. ECE Board April 1999 Angles are measured measured from the the positive horiontal a%is/ and the positive direction is counter= cloc:wise. hat are the values of sin and cos in the 4th &uadrant? A. sin , and cos cos @ , . sin @ , and cos @ , ". sin @ , and cos , #. sin , and cos ,
14. ECE Board November 2003 -he tangent of the angle of a right triangle is ,.05. hat is the csc of the angle? A. 1.0!2 . 1.!!! ". 1.660 #. 1.414 15. ECE Board November 2003 f arctan 2% * arctan !% + 45 degrees/ what is the value of %? A. 16 . 1!
21. ECE Board November 1997
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ENGINEERING MATHEMATICS I
TRIGONOMETRY
I
ind the value of % in the e&uation csc % * cot % + ! π A. 4 . ". #.
f log of 2 to the 3ase 2 plus log of % to the 3ase 2 is e&ual to 2/ then the value of % is A. 2 . =2 ". =1 #. !
π 16
π 3
26. ECE Board November 1998 $olve the e&uation cos 2 A + 1= cos2 A A. 45, D !15, . 45, D 215, ". 45, D !45, #. 45, D 245,
π 5
22. ECE Board April 1998 A man finds the angle of elevation of the top of a tower to 3e !, degrees. He wal:s 85 m nearer the tower and finds its angle of elevation to 3e 6, degrees. hat is the height of the tower? A. 0!.61 . 28 ". !, .2! #. 82.!6
20. ECE Board November 1998 "sc 52, degrees is e&ual to A. csc 2, degrees . cos 2, degrees ". tan 45 degrees #. sin 2, degrees 28. ECE Board April 1999 hat is 48,, mils e&uivalent in degrees? A. 25,, . 2!,, ". 20,, #. 22,,
2!. ECE Board April 1998 ind the angle in mils su3tended 3y a line 1, yards long at a distance of 5/,,, yards. A. 2.,4 mils . 1,.6! mils ". 1,.0! mils #. 4 mils
2. ECE Board April 1999 November 2000 "os4 A < sin4 A is e&ual to . A. cos2A . cos 4A ". sin 2A #. sin 2A
24. ECE Board April 1998 oints A and / 1,,,m apart are plotted on straight highway running Bast and est. rom A/ the 3earing of a tower " is !2 degree est of Corth and from / 3earing of " is 26 degree Corth of east. Appro%imate the shortest distance of tower " to the highway. A. !04 m. . !64 m. ". 6!6 m. #. !84 m.
!,. ECE Board April 1999 $in ( < A) is e&ual to . hen + 20,, and A is an acute angle. A. < cos A . cos A ".
25. ECE Board November 1998
!1. ECE Board April 1999
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ENGINEERING MATHEMATICS I
TRIGONOMETRY
I
f sec2 A is 52/ the &uantity 1 < sin 2 A is e&uivalent to A. ,.4, . 1.5 ". 1.5 #. 1.25
!6. ECE Board November 1999 November 2001 A central angle of 45° su3tends an arc of 12 cm. what is the radius of the circles? A. 12.!8 cm. . 15.28 cm. ". 14.28 cm. #. 11.28 cm.
!2. ECE Board November 1999 A railroad is to 3e laid < off in a circular path. hat should 3e the radius if the trac: is to 3e change direction 3y !, degrees at a distance of 150.,8m? A. !,, . 28, ". 2, #. !5,
!0. ECE Board November 1999 Fiven' y + 4cos2E. #etermine its amplitude. A. 2 . 8 ". 2 #. 4
!!. ECE Board November 1999 f (2log4 %) < (log 4) + 2/ find %. A. 12 . 15 ". 1! #. 14
!8. ECE Board April 2000 f A **" + 18,° and tan A * tan * tan " + 5.60/ find the value of tan A tan tan ". A. 5.60 . 6.15 ". 8.1! #. .12
!4. ECE Board November 1999 November 2001 f arctan (E) * arctan (1!) + π4/ the value of % is 1 A. 2 1 . 4 ". #.
!. ECE Board April 2000 -hree times the sine of a certain angle is twice of the s&uare of the cosine of the same angle. ind the angle. A. !,° . 6,° ". 45° #. 1,°
1 !
1 5
4,. ECE Board April 2001 $olve angle A of an o3li&ue triangle A"/ if a + 25/ 3 + 16 and " + 4.1 degrees. A. 52 degrees and 4, minutes . 54 degrees and !, minutes ". 5, degrees and 4, minutes #. 54 degrees and 2, minutes
!5. ECE Board November 1999 f tan4A + cot 6A/ then what is the value of angle A? A. ° . 12° ". 1,° #. 14°
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ENGINEERING MATHEMATICS I
TRIGONOMETRY
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A. ! . ! ". 2 #. 2
41. ECE Board April 2003 f tan A + 1! and cot + 2/ tan (A=) is e&ual to . A. 110 . =10 ". =11 0 #. 10
40. ECE Board April 200! Fiven' log (2% =!) + I. $olve for % if the 3ase is . A. ! . 12 ". 4 #. 5
42. ECE Board April 2003 -hree circles with radii !/ 4 and 5 inches/ respectively are tangent to each other e%ternally. ind the largest angle of a triangle formed 3y Goining the centers. A. 02.6 degrees . 05.1 degrees ". 0!.! degrees #. 0!.! degrees
48. ECE Board November 200! hat is the value of % if log (3ase %) 126 + 4? A. 5 . ! ". 6 #. 4
4!. ECE Board April 2003 ( sec A + tan A ) ind the value of if ( sec A − tan A )
4. ECE Board April 2001 f sin A + 2.5% and cos A + 5.5%/ find the value of A in degrees. A. 54.!4 . 24.44 ". !5.04 #. 45.2!
csc A + 2. A. 4 . 2 ". ! #. 1
5,. ECE Board April 2001 -riangle A" is a right triangle with right angle at ". f " + 4 and the altitude to the hypotenuse is a 1/ find the area of the triangle A". A. 2.4! . 2.,0 ". 2.11 #. 2.0,
44. ECE Board November 2003 f og 2 +%/ log ! + y/ what is log 2.4 in terms of % and y? A. !% * 2y =1 . !% * y = 1 ". !% * y *1 #. !% < y * 1 45. ECE Board November 2003 $implify the e%pression 4 cos y sin y (1 < 2 sin 2 y) A. sec 2y . cos 2y ". tan 4y #. sin 4y
51. ECE Board April 2001 -he measure of 2.25 revolutions countercloc:wise is A. 81,° . =81,° ". 8,5° #. =825°
46. ECE Board November 2003 f 2 log ! (3ase %) * log 2 (3ase %) +2 * log 6 (3ase %)/ then % e&uals .
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ENGINEERING MATHEMATICS I
TRIGONOMETRY
I
52. ECE Board November 2001 f cot 2A cot 68 + 1/ then tan A is e&ual to . A. ,.41 . 41 ". ,.14 #. 14
50. ECE Board November 1991 -he captain of a ship views the top of a lighthouse at an angle of 6, o with the horiontal at an elevation of 6 meters a3ove sea level. ive minutes later/ the same captain of the ship views the top of the same lighthouse at an angle !, , with the horiontal. Assume that the ship is moving directly away from the lighthouseD determine the speed of the ship. -he lighthouse is :nown to 3e 5, meters a3ove sea level. $olve the pro3lems 3y trigonometry. A. 4,.16 . 22.16 m min ". 1,.16 m min #. 12.16 m min
5!. ECE Board April 2002 April 1999 Assuming that the earth is a sphere whose radius is 64,, :m/ find the distance along a !=degree arc at the e&uator of the earthJs surface. A. !!!.1, :m . !!5.1, :m ". 5!!.1, :m #. !5!.,1 :m 54. ECE Board November 1998 -wo triangles have e&ual 3ases. -he altitude of one triangle is ! units more than its 3ase and the altitude of the other triangle is ! units less than its 3ase. ind the altitudes/ if the areas of the triangle differ 3y 21 s&uare units A. 4 and 1, . 4 and 26 ". 6 and 14 #. 0 and 2!
58. ECE Board April 1992 Fiven' + A sin t * cos t K + A cos t < sin t rom the given e&uation/ derive another e&uation showing the relationship 3etween / K and A and not involving any of the trigonometric function of angle t. A. P 2 - Q2 =A 2 +B 2 . P 2 +Q2 =A 2 +B2 ". P 2 - Q 2 =A 2 - B2
55. ECE Board November 1996 f sin A + 2.511%/ cos A + !.,6% and sin 2A + !.6%/ find the value of %? A. ,.265 . ,.256 ". ,.625 #. ,.214
#. P 2 +Q2 =A 2 - B2 5. ECE Board April 1993 B%press 45o in mils A. 8,,, mils . 8,, mils ". 8, mils #. 8 mils
56. ECE Board April 1991 ind the value of if it is e&ual to
sin21o +sin2 2o +sin2 3o +... +sin290o A. 45.5 . , ". infinity #. indeterminate
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TRIGONOMETRY
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6,. ECE Board April 1995 A pole cast a shadow of 15 meters long when the angle of elevation of the sun is 61°. f the pole has leaned 15° from the vertical directly toward the sun. hat is the length of the pole? A. 5!.24m . 54.2!m ". 5!.!2m #. 52.4!m
65. ECE Board November 1997 -he denominator of a certain fraction is three more than twice the numerator. f 0 is added to 3oth terms of the fraction/ the resulting fraction is ! 5. ind the original fraction A. 5 1! . ! 5 ". 4 5 #. ! 8
61. ECE Board April 2000 f A * * " +18, and tan A * tan *-an " + 5.60/ find the value of tan A tan tan " A. 5.60 . 1.08 ". 6.05 #. 1.8
66. ECE Board April 1998
21 ! ÷ Arc tan 2 cos arcsin ÷ is 2 ÷ e&ual to' A.
62. ECE Board March 1996 $olve for % in the e&uation' ∏ Arctan 2x +Arctan x = 4 A. ,.218 . ,.281 ". ,.182 #. ,.86
. ". #.
π
!
π 8 π
4
π 6
60. ECE Board April 1998 -wo electrons have speed of ,.0c and % respectively at an angle of 6,.82 degrees 3etween each other. f their relative velocity is ,.65c/ find %. A. ,.12c . ,.16c ". ,.15c #. ,.14c
6!. ECE Board April 1998 -he side of the triangle are 8/ 15/ 10 units. f each side is dou3led/ how many s&uare units will the area of the new triangle 3e? A. 21, units . 2,, units ". 18, units #. 24, units 64. ECE Board November 1997 ind the 1,,th tern of the se&uence. 1.,1/ 1.,,/ ,. L A. ,.,4 . ,.,! ". ,.,2 #. ,.,5
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ENGINEERING MATHEMATICS I
TRIGONOMETRY
I
#. 41.6 0!. "roblem# -wo sides of a triangle measures 6 cm. and 8 cm/ and their included angle is 4,°ind the third side. A. 5.2!4 cm . 4.256 cm. ". 5.144 cm. #. 5.6!2 cm.
68. ECE Board April 1998 A man finds the angle of elevation of the top of a tower to 3e !, degrees. He then wal:s 85 m nearer the tower and found its angle of elevation to 3e 6, degrees. hat is the height of the tower? A. 5.2! . 45.,1 ". 06.!1 #. 0!.61
04. "roblem# Fiven a triangle' " + 1,, °/ a + 15/ 3 + 2,. ind c' A. 20 . !4 ". 4! #. !5
6. ECE Board November 1998 f an e&uilateral triangle is circumscri3ed a3out a circle of radius 1, cm/ determine the side of the triangle. A. !4.64 cm . 64.12 cm ". !6.44 cm #. !!.51
05. "roblem# Fiven angle A+ !2 degree/ angle + 0, degree/ and side c + 20 units. $olve for side a of the triangle. A. 1,.6! units . 1, units ". 14.6! units #. 12 units
0,. "roblem# -he angle formed 3y two curves starting at a point/ called the verte%/ in a common direction. A. horn angle . inscri3ed angle ". dihedral angle #. e%terior angle
06. "roblem# n triangle A"/ angle " + 0, degreesD angle A + 45 degreesD A + 4, m. hat is the length of the median drawn from the verte% A to side "? A. !6.8 meters . !0.4 meters ". !6.! meters #. !0.1 meters
01. "roblem# -he hypotenuse of a right triangle is !4 cm. ind the length of the shortest leg if it is 14 cm shorter than the other leg. A. 16 cm . 15 cm ". 10 cm #. 18 cm
00. "roblem# -he area of the triangle whose angles are 61 °J!2JJ/ !4°14J46JJ/ and 84°!5J42JJ is 68,.6,. -he length of the 3isector of angle ". A. !5.5! . 54.!2 ". 52.4! #. 62.54
02. "roblem# A truc: travels from point M northward for !, min. then eastward for one hour/ then shifted C !, °. if the constant speed is 4,:ph/ how far directly from M/ in :m. will 3e it after 2 hours? A. 4!.5 . 40. ". 45.2
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the airplane is 0,, miles/ when will it lose contact with the carrier? A. 5 meters . 2, meters ". 1, meters #. 2.1! meters
08. "roblem# Fiven a triangle A" whose angles are A + 4,°/ + 5°and side 3 + !, cm. ind the length of the 3isector of angle ". A. 2,.45 cm . 22.!5 cm ". 21.04 cm #. 2,.85 cm.
8!. "roblem# A statue 2 meters high stands on a column that is ! meters high. An o3server in level with the top of the statue o3served that the column and the statue su3tend the same angle. How far is the o3server from the statue? A. 5 2 meters . 2, meters ". 1, meters #. 2 5 meters
0. "roblem# -he sides of a triangular lot are 1!,m/ 18,m/ and 1,m. -he lot is to 3e divided 3y a line 3isecting the longest side and drawn from the opposite verte%. -he length of this dividing line is' A. 115 meters . 1,, meters ". 125 meters #. 1!, meters
84. "roblem# rom the top of the 3uilding 1,,m high/ the angle of depression of a point A due Bast of it is !, o. rom a point due south of the 3uilding/ the angle of elevation of the top is 6, o. ind the distance A.
8,. "roblem# rom a point outside of an e&uilateral triangle/ the distance to the vertices is 1,m/ 1,m/ and 18m. ind the dimension of the triangle. A. 25.6! . 1.4 ". 45.68 #. 12.25
A. 1,,
!, !
. 100+3 30 ".
100- 3 30
30 #. 100+ 62
81. "roblem# oints A and 1,,,m apart are plotted on a straight highway running Bast and est. rom A/ the 3earing of a tower " is !2 degrees C of from the 3earing of " is 26 degrees C of B. appro%imate the shortest distance of tower " to the highway. A. 264 meters . 204 meters ". 284 meters #. 24 meters
85. "roblem# An o3server found the angle of elevation at the top of the tree to 3e 20o. After moving 1,m closer (on the same vertical and horiontal plane as the tree)/ the angle of elevation 3ecomes 54o. ind the height of the tree. A. 8., meters . 0.5! meters ". 8.25 meters #. 0.,2 meters
82. "roblem# An airplane leaves an aircraft carrier and flies $outh at !5, mph. the carrier travels $ !,o B at 25 mph. if the wireless communication range of
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ENGINEERING MATHEMATICS I
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86. "roblem# rom point A at the foot of the mountain/ the angle of elevation of the top is 6,°. After ascending the mountain one (1) mile at an inclination of !,°to the horion/ and reaching a point "/ an o3server finds that the angle A" is 1!5. -he height of the mountain in feet is' A. 14/86 . 12/4! ". 14/08 #. 12/225
at the top of the tower from A and / which are 5, ft. apart/ at the same elevation on a direct line with the tower. -he vertical angle at a point A is !,°and at a point is 4,. hat is the height of the tower? A. 11,.2 feet . 2.54 feet ". 14!.0 feet #. 85.25 feet
80. "roblem# A 5, meter vertical tower casts a 62.! meter shadow when the angle of elevation of the sum is 41.6 °. the inclination of the ground is' A. 4.!!° . 4.02° ". 5.6!° #. 5.10°
A. 28,
1. "roblem# ind the supplement of an angle whose compliment is 62°
. 152, ". =2 6.20 #.
1 2
2. "roblem# A certain angle has a supplement five times its compliment. ind the angle. A. 60.5, . 150.5, ". 168.5, #. 186.5,
88. "roblem# A vertical pole is 1,m from a 3uilding. hen the angle of elevation of the sun is 45 °the pole cast a shadow on the 3uilding 1m high. ind the height of the pole. A. 12meters . 11 meters ". , meter #. 1!meters
!. "roblem# -he sum of the interior angles of the triangle is e&ual to the third angle and the difference of the two angles is e&ual to 2! of the third angle. ind the third angle. A. 45, . 05, ". !, #. ,,
8. "roblem# A pole cast a shadow of 15 meters long when the angle of elevation of the sun is 61 . if the pole has leaned o
15°from the vertical directly toward the sun/ what is the length of the pole? A. 52.4!meters . 54.2! meters ". 52.25 meters #. 5!.24 meters
4. "roblem# -he measure of 1
1
revolutions 2 counter=cloc:wise is' A. 54,, . 52,, ". 58,, #. 55,
,. "roblem# An o3server wishes to determine the height of the tower. He ta:es sights
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ENGINEERING MATHEMATICS I
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5. "roblem# -he measure of 2.25 revolutions countercloc:wise is' A. 8,, degrees . 82, degrees ". 81, degrees #. 85, degrees
1,,. "roblem# -he insides of a right triangle are in arithmetic progression whose common difference is 6 cm. ts area is' A. 20, cm2 . !4, cm2 ". 216 cm2 #. 144 cm2
6. "roblem# f -an θ + %2/ which of the following is correct? 1 cos! = A. 1+x4 1 sin! = . 1+x4 1 ". csc! = 1+x4 1 #. tan! = 1+x4
1,1. "roblem# rom the top of tower A/ the angle of elevation of the top of the tower is 46° . rom the foot of a tower the angle of elevation of the top of tower A is 28°. oth towers are on a level ground. f the height of tower is 12, m./ How far is A from the 3uilding? A. 42.! m. . 4,.0 m. ". !8.6 m. #. 44.1 m.
0. "roblem# n an isosceles right triangle/ the hypotenuse is how much longer than its sides? A. 2 times . 2 times ". 1.5 tines #. none of these
1,2. "roblem# oint A and are 1,, m apart and are on the same elevation as the foot of the 3uilding. -he angles of elevation of the top of the 3uilding from point A and are 21 ° and !2°/ respectively. How far is A from the 3uilding? A. 265.4 m. . 200. m. ". 25.2 m. #. 25.2 m.
8. "roblem# ind the angle in mils su3tended 3y a line 1, yards long at a distance of 5/,,, yards. A. 1 mil . 2 mils ". 6 mils #. ! mils
1,!. "roblem# A man finds the angle of elevation of the top of a tower to 3e !, degrees. He wal:s 85 m. nearer the tower and finds its angle of elevation to 3e 6, degrees. hat is the height of the tower? A. 0!.61 . 06.!1 ". 0!.!1 #. 01.!6
. "roblem# -he angle or inclination of ascends of a road having 8.25N grade is degrees. A. 5.12 degrees . 1.86 degreed ". 4.02 degrees #. 4.20 degrees
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ENGINEERING MATHEMATICS I
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". 2 2 #. 6
1,4. "roblem# -he angle of elevation of point " from point is 2 °42JD the angle of elevation of " from another point A !1.2 m. directly 3elow is 5 °2!J. How high is " from the horiontal line through A? A. !5.1 meters . 52.! meters ". 40.1 meters #. 66. meters
1,8. "roblem# A cloc: has a dial face 12 inches in radius. -he minute hand is inches long while the hour hand is 6 inches long. -he plane of rotation of the minute hand is 6 inches long. -he plane rotation of the minute hand is 2 inches a3ove the plane of rotation of the hour hand. ind the distance 3etween the tips of the hands at 5'4, AM. A. 8.2! in. . 1,.65 in. ". .10 in. #. 11.25 in.
1,5. "roblem# A rectangle piece of land 4, m % !, m is to 3e crossed diagonally 3y a 1,=m wide roadway as shown. f the land cost 1/5,,.,, per s&uare meter/ the cost of the roadway is' A. 4,1.1, . 6,/165.,, ". 651/5,,.,, #. 6,1/65,.,,
1,. "roblem# f the 3earing of A from is $ 4, °/ then the 3earing of from A is' A. $ 4,° . $ 5,° ". C 4,°B #. C 05°
1,6. "roblem# 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.! m. ith the longer side lying horiontally on the ground/ he props up the other corner of the triangle with a vertical pole ,. m long. hat would 3e the area of the shadow on the ground when the sun is vertically overhead? A. ,.5m2 . ,.05m2 ". ,.84m2 #. ,.5 m2
11,. "roblem# A plane hillside is inclined at an angle of 28 degree with the horiontal. A man wearing s:is can clim3 this hillside 3y following a straight path inclined at an angle of 12 degree to the horiontal/ 3ut one without s:is must follow a path inclined at an angle of 5 degree with the horiontal. ind the angle 3etween the directions of the two paths. A. 1,.24 degree . 1!.21 degree ". 10.22 degree #. 15.56 degree
1,0. "roblem# A rectangular piece of wood 4 % 12 cm tall is tilted at an angle of 45 degrees. ind the vertical distance 3etween the lower corner and the upper corner. A. 4 2 . 8
2
111. "roblem# -he sides of the triangle A" are A + 15 cm/ " + 18 cm/ "A + 24 cm. ind the distance from the point of intersection of the angle 3isectors to side A.
2
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ENGINEERING MATHEMATICS I
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A. 5.45 . 5.!4 ". 4.0! #. 6.25 112. "roblem# -wo straight roads intersect to form an angle of 05 degrees. ind the shortest distance from one road to a gas station on the other road 1 :m./ from the Gunction. A. 1.241 . 4.0!2 ". 2.241 #. !.0!2
A. 45/,65/046., . 56/406/,62.,0 ". 64/054/,!4.,2 #. 24/245/258.,, 110. "roblem# f the Freenwich Mean -ime (FM-) is 0 A.M. hat is the time in a place located at 1!5 degrees east longitude? A. 4 .M. . 6 .M. ". 2 .M. #. 5 .M. 118. "roblem# f Freenwich Mean -ime (FM-) is a.m. what is the time in a place 45o of longitude? A. 6 A.M. . 4 A.M. ". 2 A.M. #. 8 A.M.
11!. "roblem# A train travels 2.5 miles up on a straight trac: with a grade of 1 °1,J. hat is the vertical rise of the train in that distance? A. ,.016 miles . ,.,51 miles ". ,.20 miles #. ,.,45 miles
11. "roblem# ind the distance in nautical miles and the time difference 3etween -o:yo and Manila if the geographical coordinates of -o:yo and Manila are ( !5.65 degree north at.D 1!. 05 degrees Bast long.) and ( 14.58 degree CorthD 12,. 8 degree long.)/ respectively. A. 146 nautical milesD 4.25 hrs. . 2615 nautical milesD 1.52 hrs. ". 1612 nautical milesD 1.25 hrs. #. 1485 nautical milesD 1.25 hrs.
114. "roblem# our holes are to 3e spaced regularly on a circle of radius 2, cm. ind the distance OdP 3etween the centers of the two successive holes. A. 2, 2 . 1,
!
". 15 2 #. 5
2
115. "roblem# n a spherical triangle A"/ A + 116°1J/ + 55°!,J and " + 8,°!0J. ind the value of side a. A. 115.50 degree . 11,.56 degree ". 118.10 degree #. 112.12 degree
12,. "roblem# An isosceles spherical triangle has angle A++ 54 degree and side 3+ 82 degrees. ind the measure of the third angle. A. 158°24J15P . 155°24J15P ". 168°24J15P #. 165°24J15P
116. "roblem# "onsidering the earth as a sphere of radius 6/4,, :m. ind the area of a spherical triangle on the surface of the of the earth whose angels are 5,°8J and 12, °.
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ENGINEERING MATHEMATICS I
TRIGONOMETRY
I
An o3server m. horiontally away from the tower o3serves its angle of elevation to 3e 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 . 4 m ". ! m #. 5 m
121. "roblem# 7n one side of a paved oath wal: is a flag staff on top of it. -he pedestal is 2 m. in height whole the flag staff is ! m/ high. At the opposite edge of the path wal: the pedestal and flag staff su3tends e&ual angles. "ompute the width of the path wal:. A. 4.40 m . 5.!4 m ". 5.04 m #. !.08 m
126. "roblem# A man finds the angle of elevation of the top of a tower to 3e !, degrees. He wal:s 85 m. nearer the tower and finds its angle of elevations to 3e 6, degrees. hat is the height of the tower? A. 0!.61 m . 45.!6 m ". 66.!6 m #. 54.21 m
122. "roblem# $implify the e&uation sin 2 θ (1 * cot 2 θ) A. $in 2 θ . 1 ". $in 2 θ $ec 2 θ #. "os 2 θ
120. "roblem# oints A and are 1,, m. Apart and are of the same elevation as the foot of the 3ldg. -he angles of elevation of the top of the 3ldg. from points A and are 21 degrees and !2 degrees respectively. How ar is A from the 3ldg. in meters? A. 200.!6 . 201.62 ". 265.42 #. 25.28
12!. "roblem# -he angle of elevation of a top of a tree from a point 1, m. horiontally away from the tree is twice the angle of elevation at a point 5, m. from it. ind the height of the tree. A. !4.25 . 20.8 ". 46.58 #. !8.0!
128. "roblem# A and are summits of two mountains rise from a horiontal plain. 3eing 12,, m a3ove the plain. ind the height of A/ it 3eing given that its angle of elevation as seen from a point " in the plane (in the same vertical plane with A and ) is 5, degree/ while the angle of depression of " viewed from is 28°58J and the angle su3tended at 3y A" is 5, degree. A. !2,,.2, m . !,,2.!! m ". 28.42 m
124. "roblem# A vertical pole consists of two parts/ each one half of the whole pole. At a point in the horiontal plane which passes through the foot of the pole and !6 m. rom it/ the upper half of the pole su3tends an angle whose tangent is 1 !. How high is the pole? A. 04 m or !6 m . 02 m and !6 m ". 6, m or !, m #. 8, m or 4, m 125. "roblem#
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ENGINEERING MATHEMATICS I
TRIGONOMETRY
I
#. 2840.64 m
our hours earlier a freight ship started from the same point at the speed of 8 :ph with a direction C 15°42J . #etermine the num3er of hours it will ta:e the freight ship to 3e e%actly C 05°25J of the passenger ship. A. 2 . 4 ". 8 #. 5
12. "roblem# A 4, m high tower stands vertically on a hillside (sloping ground) which ma:es an angle of 18 degree with the horiontal. A tree also stands vertically up the hill from the tower. An o3server on the top of the tower finds the angle of depression of the top of the tree to 3e 26 degrees and the 3ottom of the tree to 3e !8 degree. ind the height of the tree. A. 15.2 m . 1,.62 m ". 0.!8 m #. 1!.20 m
1!!. "roblem# A truc: travels from point M north ward for !, min./ then eastward for one hour/ then shifted C. !, degrees est. f the constant speed is 4, :ph/ how far directly from M in :m/ will it 3e after 2 hours? A. 45.22 . 40.88 ". 41.66 #. 4!.55
1!,. "roblem# -wo towers A and stands 42/ apart on a horiontal plane. A man standing successively at their 3ases o3serves that the angle of elevation of the top of tower is twice that of the 3ases the angles of elevation are complimentary. ind the angle of elevation of the tower from the 3ase of tower A if the height of the tower is !, m. A. 28.!2 degree . !8.58 degree ". 4,.20 degree #. !8.58 degr ee
1!4. "roblem# A car travels northward from a point for one hour/ then eastward for !, one hour/ then eastward for !, min then shifted C !,° B. After e%actly 2 hours/ the car will 3e 64.0 :m directly away from . hat is the speed of the car in 9ph? A. 45 . 4, ". 5, #. 55
1!1. "roblem# A ship started sailing $ 42°!5J at the rate of 5 :ph. After 2 hours/ ship started at the same port going C 46°2,J at the rate of 0 :ph. After how many hours will the second ship 3e e%actly north of ship A? A. 4.,! . !.84 ". 2.6 #. 5.8
1!5. "roblem# A motorcycle travels northward from point for half an hour/ then eastward for one hour/ then shifted C !,° . After e%actly 2 hours/ the motorcycle will 3e 40.88 :m. Away from . hat is the speed of the motorcycle in :ph? A. 45 . 4! ". 1!, #. !!
1!2. "roblem# A passenger ship sailed northward with a direction C 42 °25J B at 14 :ph.
1!6. "roblem#
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ENGINEERING MATHEMATICS I
TRIGONOMETRY
I
A 3oat can travel 8 mi hr. in still water. hat is its velocity with respect to the shore if it heads 55 degrees east of north in a current that moves ! mihr west? A. 5.4 mph . 6.04! mph ". 4.556 mph #. 8.6! mph
-he area of an isosceles triangle is !6 m2 with the smallest angle e&ual to one third of the other angle. ind the length of the shortest side. A. 5.0! m. . .22 m. ". 8.46 m. #. 12.88m. 142. "roblem# -he difference 3etween the angles at the 3ase of a triangle is 10 ° 48J and the sides su3tending this angles are 1,5.25 m and 06.05 m. ind the angle included 3etween the given sides. A. , degrees . 8, degrees ". 0, degrees #. 6, degrees
1!0. "roblem# f the figure/ # and #" are angle 3isectors. f angle A+ 8,°/ how many degrees is angle A#"? A. 14, degrees . 12, degrees ". 15, degrees #. 1!, degrees 1!8. "roblem# if a triangle A"/ angle A + 6, degrees and the angle +45 degrees/ what is the ratio of sides " to side A"? A. 1.!6 . 1.22 ". 1.48 #. 1.1
14!. "roblem# -hree forces 2, C/ !, C and 4, C are in e&uili3rium. ind the angle 3etween the !, C and 4, C forces. A. 25.0 degrees .4, degrees ". 28.6 degrees #. !,°15J25P
1!. "roblem# ind the angle of a triangle if a + 1!2 m./ 3 + 224 m. And " + 28.0 degrees. A. 5°,!J25P . 4°,!J25P ". !°,!J25P #. 1°,!J25 ”
144. "roblem# 7ne leg of the right triangle is 2, units and the hypotenuse is 1, units longer than the other leg. ind the lengths of the hypotenuse. A. 2, . 25 ". 1, #. 15
14,. "roblem# -he area of an isosceles triangle is 02 s&.m. f the two e&ual sides ma:e an angle of 2, degree with the third side/ compute the length of the longest side. A. 25.62 . 20.84 ". !1.22 #. 28.1!
145. "roblem# #etermine the sum of the positive valued solution to the simultaneous e&uations' %y + 15/ y + !5/ % + 21. A. 15 . 1! ". 10 #. 1
141. "roblem#
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ENGINEERING MATHEMATICS I
TRIGONOMETRY
I
the sum is 2, degrees and its 3earing is $ 6,°B/ calculate the area of the shadow of the wall on the horiontal ground. "# is on the ground portion of the wall and has a direction of due north. A. 08.5! s&. m. . 8,.!, s&. m. ". 0!.45 s&. m. #. 01.4, s&. m.
146. "roblem# A rectangle A"# which measures 18 % 24 units is folded once/ perpendicular to diagonal A"/ so that the opposite vertices A and " coincide. ind the length of the fold. A. 18.05 cm. . 22.5 cm. ". 21.5 cm. #. 1.5 cm.
151. "roblem# A spherical triangle A" has an angle " + , degrees and sides a + 5, degrees and c + 8, degrees. ind the value of O3P in degrees? A. 05.44 . 04.!! ". 06.55 #. 0!.22
140. "roblem# A#B is a s&uare section and #" is an e&uilateral triangle with " outside the s&uare. "ompute the value of angle A"B. A. !5 degree . 5, degree ". !, degree #. 4, degree
152. "roblem# A point 7 is inside a s&uare lot/ if the distances from point 7 to the three successive corners of the s&uare lot are 5 m./ ! m. And 4 m/ respectively/ find the area of the s&uare lot. A. !2.1 . 2!.2 ". 45.4 #. !6.6
148. "roblem# ind the sum of the interior angles of the vertices of a five pointed star inscri3ed in a circle. A. 12, degrees . 14, degrees ". 18, degrees #. 10, degrees
15!. "roblem# ind the angle in mils su3tended 3y a line 1, yards long at a distance of 5/,,, yards. A. 1., yards . 2.,4 mils ". 1.5 yards #. 1.! yards
14. "roblem# A s&uare section A"# has one of its sides e&ual to %. oint B is inside the s&uare forming an e&uilateral triangle B" having one side e&ual to the side of a s&uare. t is re&uired to compute the angle of AB#. A. 15, degrees . 14, degrees ". 15, degrees #. 12, degrees
154. "roblem# -he A point is inside an e&uilateral triangle/ if the distances from to the three vertices of the triangle are ! m/ 4 m/ and 5 m/ respectively/ ind the area of the triangle A. 10.5 . 12.2! ". 1.85 #. 2,.,
15,. "roblem# A"# is a vertical wall A# + ! m. High/ A + 1, m. long. -he wall is 3uilt on a north south line on a horiontal ground. f the elevation of
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ENGINEERING MATHEMATICS I
TRIGONOMETRY
I
-he corners of a triangle lot are mar:ed 1/ 2/ and ! respectively. -he length of side ! < 1 is e&ual to 5,, m. -he angles 1/ 2 and ! are 6, degrees/ 8, degrees and 4, degrees respectively. f the area of 5/!52 s&. m. is cut off on the side ! < 1 such that the dividing line 4 =5 is parallel to ! < 1. 1. "ompute the length of line 4 =5. 2. "ompute the area of 2 < 4 < 5. !. "ompute the distance 2 < 4.
155. "roblem# f the sides of the triangle are 2% * !/ %2 * 2%/ find the greatest angle. A. 12, degrees . 1,, degrees ". 11, degrees #. 1!, degrees 156. "roblem# ind the value of θ in the e&uation "osh 2 % < $inh 2 θ +2cosθ. A. 45 degrees . 6, degrees ". !, degrees #. !5.6 degrees
161. "roblem# hat are the e%act values of the cosine and tangent trigonometric functions of acute angle A/ Fiven that $in A +
150. "roblem# $olve for % if "osh % * $inh % + 0.!8 A. 2 . 1, ". 1 #. e%
A.
2
! 0
.
8 5
. ".
158. "roblem# ind % if "osh 2 % < $inh 2 % * tanh 2 % * sech 2 %+ "osh % * $inh % A. 1 . ln 2 ". e% #. %=%
#.
3 10 20 4 10 15 3 8 10
162. "roblem# $implify the e%pression sec θ = (sec θ) sin 2 θ A. "os 2 θ . "os θ ". $in θ #. $in 2 θ
15. "roblem# -riangle EQR has 3ase angles E + 52 degrees and R + 6, degrees distance ER + 4,, m long. A line A which is 2, m long is laid out parallel to ER. 1. "ompute the area of triangle EQR. 2. "ompute the area of AER. !. -he area of AQ is to 3e divided into two e&ual parts. "ompute the length of the dividing line which is parallel to A.
16!. "roblem# A flagpole is places on top of the pedestal at a distance of 15 m from the o3server. -he height of the pedestal is 2,m. f the angle su3tended 3y the flagpole at the o3server is 1, degrees. 1. "ompute the angle of elevation of the flagpole. 2. "ompute the height of the flagpole.
16,. "roblem#
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ENGINEERING MATHEMATICS I
TRIGONOMETRY
I
!. f the o3server moves a distance of 5m. towards the pedestal/ what would 3e the angle of pedestal/ what would 3e the angle of elevation of the flagpole at this pt.
". 2 mils #. 0 mils 16. "roblem# n an isosceles right triangle/ the hypotenuse is how much longer than its sides? A. 2 times . 2 times ". 1.5 times #. none of these
164. "roblem# A right spherical triangle has an angle " + , degrees/ a + 5, degrees/ and c + 8, degrees. ind the side 3. A. 08.66 degrees . 45.!! degrees ". 05.8 degrees #. 04.!! degrees
10,. "roblem# f tan θ + %2/ which of the following is incorrect? 1 A. sec θ + 1 + E4 1 . cos θ + 1 + E4 1 ". sin θ + 1 + E4 1 #. cot θ + 1 + E4
165. "roblem# "alculate the area of a spherical triangle whose radius is 5 m and whose angles are 4, degrees/ 65 degrees/ and 11, degrees. A. 15. 20 . 10.2! ". 21.21 #. 15.80 166. "roblem# -he sides of right triangle are in arithmetic progression whose common difference if 6 cm. ts area is' A. 20, cm2 . 216 cm2 ". 14, cm2 #. 16, cm2
101. "roblem# ind the value of y' y + (1 * cos 2θ) tan θ. A. sin θ . sin 2θ ". cos ,θ #. tan θ 102. "roblem#
160. "roblem# -he angle of inclination of ascends of a road having 8.25 N grade is degrees. A. 4.02 . 5.12 ". 4.20 #. 1.86
$implifying the e&uation sin cot 2 θ) gives' A. ,.5 . eg ". 1 #. e2E
2
θ (1 *
10!. "roblem# hich of the following e%pression in e&uivalent to sin 2θ A. 2tanθcotθ . 2sinθcosθ ". 2sinθ #. cotθ
168. "roblem# ind the angle in mils su3tended 3y a line 1, yards long at a distance of 5/,,, yards. A. 4 mils . 2.5 mils
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ENGINEERING MATHEMATICS I
TRIGONOMETRY
I
18,. "roblem# -he is the line or line segment that divides the angle into two e&ual parts. A. angle 3isector . apothem ". perpendicular 3isector #. terminal side
104. "roblem# -he e&uation 2 sinh % cosh % is e&ual to' A. e=% . e% ". cosh % #. sinh 2% 105. "roblem# ind the value of sin (, °*A) A. cos A sin A. . < cos A ". cos A #.
181. "roblem# are the lines 3isecting the angles formed 3y the sides of the triangles and their e%tensions. A. B%secants . nternal angle 3isector ". erpendicular 3isector #. B%terior angle 3isectors
106. "roblem# f sin(% * y) + ,.066 and sin (% < y) +,.10!6. ind sin % cos y. A. ,.65 . ,.60!2 ". ,.468 #. ,.856!
182. "roblem# -he two legs of the triangle are !,, and 15, each/ respectively. -he angle opposite the 15, m side is 26 degree. hat is the third side? A. !41.08 m . 218.61 m ". 282.15 m #. 105.2! m
100. "roblem# $olve for θ if coth2% < csch2 % + e%secant θ. A. 45° . !,° ". 6,° #. 2,°
18!. "roblem# -he sides of the triangular lot are 1!,/ 18, and 18, m. -he lot is to 3e divided 3y a line 3isecting the longest side and drawn from the opposite verte%. ind the length of the line. A. 12, . 24, ". 125 #. 2,,
108. "roblem# A transformation consisting of a constant offset with no rotation or distortion. A. screw . translation ". reflection #. torsion
184. "roblem# -he sides of the triangle are 15/ 150 and 21,/ respectively. hat is the area of the triangle? A. 14/ 586.2 . 28/ 586 ". 16/ 586.2 #. 41/ 586.2
10. "roblem# An angle whose endpoints are located on a circleJs circumference and verte% located at the circleJs center A. central angle . e%terior angle ". supplementary angle #. complementary angle
185. "roblem#
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ENGINEERING MATHEMATICS I
TRIGONOMETRY
I
3 f sin A = and sin A +B =1/ find 5 cos ' A. ,.0, . ,.8, ". ,.6, #. ,.1,,
-he turning of an o3Gect or coordinate system 3y an angle a3out a fi%ed point. A. involution . revolution ". dilation #. rotation
186. "roblem# A Meralco tower and a monument stand on a level plane. -he angles of depression of the top and 3ottom of the monument viewed from the top of the Meralco tower at 1! degrees and !5 degrees respectively. -he height of the tower is 5, m. ind the height of the monument. A. !!.541 m . 64.12 m ". !2.1, #. !6.44 m
1,. "roblem# A rotation com3ined with an e%pansion or geometric con = traction. A. screw . shift ". twirl #. twist 11. "roblem# ind the sin % if 2sin% * !cos% = 2 + , A. 1 and 5 1! . ! and 5 15 ". 15 and 5 1! #. 1 and =5 1!
180. "roblem# A wire supporting a pole is fastened to it 2, feet from the ground and to the ground 15 feet from the pole. #etermine the length of the wire and the angle it ma:es with the pole. A. 25 ft/ !6.80 degrees . 25 ft/ 5!.10 degrees ". 24 ft/ !6.80 degrees #. 24 ft/ 5!.10 degrees
12. "roblem# f sin A + 4 5/ A in &uadrant / sin + 0 25/ in &uadrant / find the sin (A * ) A. 2 5 . ! 5 ". 4 5 #. 6 5
188. "roblem# oint A and are 1,, m apart and are of the same elevation as the foot of a 3uilding. -he angles of elevation of the top of the 3uilding from points A and are 21 degrees and !2 degrees respectively. How far is A from the 3uilding in meters? A. 25.28 . !25.,, ". 454.85 #. 512.,5
1!. "roblem# f sin A + 2.50 %D cos A + !.,6%/ and sin 2A + !.!%/ find the value of the %. A. ,.15, . ,.1,, ". ,.25, #. ,.!5, 14. "roblem# if cos θ +
18. "roblem#
! 2 / then find the value
of % if % + 1= tan 2 θ? A. 2 !
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ENGINEERING MATHEMATICS I
TRIGONOMETRY
I
. 4 ! ". 8 #. 1 15. "roblem# f sin θ = cos θ + = 1!/ what is the value of sin 2θ? A. 8 . 4 ". 1, 12 #. 1, 15
A. 41°14J48P/ 6!°1,P48P . 45°2!J4!P/ 66°12J45P ". 4,°1!J!5P/ 66°12J45P #. 22°24J!P/ 6,°12J45P
16. "roblem# Fiven the parts of the spherical triangle' A + 6,°!,J 3 + !8°15J a + 4,°!,J
18. "roblem# rom the given parts of the spherical triangle A"/ compute for the angle A. A + 52°!,J + 48°!4J " + 12,°15J
A. 45°!5J . 44°!,J ". 40°4!J #. 46°4,J 10. "roblem# n the spherical triangle shown/ following parts are given' A + 4,°18J " + 05°,,J c + 1,,°1,J 3 + 65°25J
A. 128°42J . 50°16J ". 141°!2J #. 114°16J 1. "roblem# -he is an imaginary rotating sphere of gigantic radius with the earth located at its center. A. celestial sphere . chronosphere ". e%osphere #. astrological sphere 2,,. "roblem#
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ENGINEERING MATHEMATICS I
TRIGONOMETRY
I
-wo celestial coordinate are' A. right ascension and declination . longitude and latitude ". Corth ole and $outh ole #. enith and nadir
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ENGINEERING MATHEMATICS I
