A.6. FORMACIÓN DEL PERSONAL DE CONTROL OFICIAL
A.6.2. Formación Estatal
D. The distance of a point from a line.
993. Which of the following systems of angle measurements uses the degree as the unit of measure?
A. circular system C. sexagesimal system
B. mil system D. grade system
994. In what quadrant will angle A terminate if sec A is positive and csc A is negative.
A. I B. II C. III D. IV
995. Which of the following relations is not true ?
A. sinx = (tanx/secx) C. cotx = cscx cosx
B. (cotx/cscx) = (sinx/tanx) D. (secx/tanx) = (cosx/cotx)
996. Within what limits between between 0 degrees and 360 degrees must the angle θ lie if cos θ
= -2/5 ?
A. between 0 degrees and 180 degrees B. between 90 degrees and 180 degrees C. between 90 degrees and 270 degrees D. between 90 degrees and 360 degrees
997. The coreference angle of any angle A is the positive acute angle determined by the terminal side of A and the y-axis. What is the coreference angle of 290 degrees ?
A. 70 degrees B. 50 degrees C. 30 degrees D. 20 degrees
998. A measure of 3200 mils is equal to A. 90 deg
B. 45 deg C. 180 deg D. 120 deg
999. The value of vers θ is equal to A. 1 - cosθ
B. 1 - sinθ C. 1 + cosθ D. 1 +sinθ
1000. To find the interior angles of a triangle whose sides are given, use the law of A. sine B. cosine C. tangent D. secant
1001. The point P(x,y) where x 0 and y > 0 is located in quadrant A. I or IV B. II or III C. I or II D. III or IV
1002. Which of the following relations is true for any angle θ ? A. sin(-θ) = sin θ C. tan(-θ) = tan θ B. sec(-θ) = sec θ D. csc(-θ) = csc θ 1003. Coversine A is equal to
A. 1 - cosA C. 1 + cosA
B. 1 - sin A D. 1 + sin A
1004. The terminal side of -1,500 degrees will lie in quadrant
A. one B. two C. three D. four
1005. Which of the following is false as the angle A increases from 0 degrees to 90 degrees ? A. sin A increases from zero to one
B. tan A increases from zero to infinity C. cos A decreases from one to zero D. sec A decreases from one to infinity
1006. Which of the following functions is positive if angle A terminates in the second quadrant ?
A. csc A B. tan A C. sec A D. cos A
1007. An angle in standard position and whose terminal side falls along one of the coordinate axes is called a
A. reference angle C. quadrantal angle B. vertical angle D. central angle
1008. Which of the following pairs of angles in standard positions are coterminal angles ? A. 710 degrees and -10 degrees
B. 120 degrees and 60 degrees C. -240 degrees and 30 degrees D. 325 degrees and -40 degrees
1009. The gradient of the line in the figure is A. tan θ
B. -1/tan θ C. -tan θ D. cot θ
1010. Which of the following is true in quadrants III and IV ? A. negative cosecant C. negative cotangent B. positive sine D. positive tangent 1011. Which of the following is not a first quadrant angle ?
A. 450 degrees C. -330 degrees
B. 60 degrees D. -120 degrees
1012. If tan θ > 0 and cosθ < 0, then θ is a
A. first quadrant angle C. third quadrant angle B. second quadrant angle D. fourth quadrant angle
1013. If an angle is in the standard position and its measure is 215 degrees, the its reference angle is
A. 25 degrees B. 30 degrees C. 35 degrees D. 40 degrees 1014. In the second quadrant, which of the following is true ?
A. The tangent and secant are positive B. The sine and cosecant are positive C. The cotangent and cosecant are positive D. The sine and tangent are positive
1015. In what quadrant can we locate the point (x, -4) if x is positive ?
A. I B. II C. III D. IV
1016. In what quadrants do the secant and cosecant of an angle have the same algebraic sign?
A. II and IV B. I and II C. I and III D. III and IV 1017. If cos 3A + sin A = 0, find the value of A.
A. 30 degrees B. 45 degrees C. 60 degrees D. 90 degrees 1018. If tan A = 2 and tan B = 1/2, find A + B.
A. 90 degrees B. 30 degrees C. 45 degrees D. 60 degrees 1019. If sin x = 5/13 , find sin 2x.
A. 120/169 B. 25/169 C. 10/13 D. 12/13
1020. If cot θ = square root of 3 and cos θ < 0, find csc θ.
A. 2 B. -2 C. 1/2 D. -1/2
1021. If sin A = -5/13 and A in quadrant III, find cot A.
A. 12/5 B. -12/5 C. 5/12 D. -5/12
1022. Find the value of sin(Arecos 15/17).
A. 8/9 B. 8/2 C. 17/19 D. 8/17
1023. The cosecant of 960 degrees is equal to A. -2( square root of 3 / 3)
B. 2( square root of 3 / 3) C. 1/2
D. -1/2
1024. If sin 3A = cos 6B, then A. A - 2B = 90 degrees B. A + 2B = 90 degrees C. A + B = 180 degrees D. A + 2B = 30 degrees
1025. What is the period of y = 3 sin(x/2) ?
A. 2 pi B. 3 pi C. 4 pi D. 6 pi
1026. If the product of cot 2θand cot 68 degrees is equal to unity, find θ.
A. 13 degrees B. 12 degrees C. 11 degrees D. 10 degrees
1027. Sec A - cos A is identically equal to A. sin A cot A
B. cos A tan A C. sin A tan A D. cos A cot A
1028. Simplify ( sin θ/ 1 - cos θ) - ( 1 + cos θ/ sin θ)
A. sin²θ B. cos²θ C. 1 D. 0
1029. If tan x = 1/2 and tan y = 1/3, find tan (x + y).
A. 1 B. 2/3 C. 2 D. 1/2
1030. If cos θ= 3/5 and θ in quadrant IV, find cos2θ
A. 7/25 B. -7/25 C. 24/25 D. -24/25
1031. simplify (sinθ + cosθtanθ)/(cosθ)
A. tanθ B. 2cotθ C. sinθ D. 2tanθ
1032. If Arcsin(2x) = 30 degrees, find x.
A. 0.20 B. 0.25 C. 0.3 D. 0.35
1033. If sin 40 degrees + sin 20 degrees = sin θ, find the value of θ.
A. 20 degrees B. 60 degrees C. 80 degrees D.120 degrees 1034. The angle that is equal to one half of its supplement is
A. 60 degrees B. 90 degrees C. 80 degrees D. 45 degrees 1035. Find the equivalent value of y in the equation y = (1 + cos 2θ) / (cot θ)
A. sin2θ B. cos2θ C. sinθ D. cosθ
1036. If tan A = -3 and tan B = 2/3, find tan(A - B).
A. -11/9 B. -10/9 C. -13/9 D. -12/9
1037. If cos 65 degrees + cos 55 degrees = cos θ, find the θ in radians.
A. 1.832 B. 1.658 C. 0.7853 D. 0.0873
1038. If tan (A / 4) = cot A, find A.
A. 52 degrees B. 72 degrees C. 42 degrees D. 62 degrees 1039. Simplify cos^4 x- sin^4 x
A. cos 4x B. sin 4x C. sin 2x D. cos 2x
1040. If tan 4x = cot 6y, then
A. 2x - 3y = 45 degrees C. 4x - 6y = 90 degrees B. 2x + 3y = 45 degrees D. 6y - 4x = 90 degrees 1041. Simplify Arctan(1/3) + Arctan(1/5)
A. Arctan (7/4) C. Arctan (8/15)
B. Arctan (4/7) D. Arctan (1/15)
1042. If sin A =3.5x and cos A = 5.5x, find angle A.
A. 32.47 degrees C. 34.47 degrees
B. 33.47 degrees D. 35.47 degrees
1043. If the tangent of an angle x is 3/4, find the value of the cosine of 2x.
A. 0.60 B. 0.28 C. 0.8 D. 0.38
1044. Find the angle which a 9-m ladder will make with the ground if it is leaned against a window still 6m high.
A. 21.8 degrees C. 41.8 degrees
B. 31.8 degrees D. 51.8 degrees
1045. The expression (1 -sinx) / (cosx) is equal to
A.tanx B.1 C.(1 - cosx)/sinx D.(cosx) / (1 + sinx)
1046. A tree 30 m long casts a shadow 36 m long. Find the angle of elevation of the sun.
A. 39.41 degrees C. 39.81 degrees
B. 39.51 degrees D. 39.61 degrees
1047. Which of the following is true ?
A. tan(180 degrees + θ) = - tanθ C. tan(90 degrees + θ) = -tanθ B. tan(180 degrees - θ) = -tan θ D. tan(270 degrees - θ) = - tanθ 1048. Express 3i + 5 + (square root of -16) in the standard form.
A. 5 - 7i B. 5 + 7i C. -5 + 7i D. -5 - 7i
1049. Write (square root of 2) cis 135 degrees in rectangular form.
A. 1 -i B. -1 + i C. -1 - i D. 1 + i
1050. Give the conjugate of 2 + (square root of -25) in the standard form.
A. 2 - 5i B. 2 + 5i C. -2 + 5i D. -2 -5i
1051. For the trigometric function y = a sin(bx +c), the absolute value of the ratio c/b is called A. amplitude B. period C. argument D. phase shift
1052. If sin2x sin4x = cos2x cos4x, find the value of x.
A. 13° B. 14° C. 15° D. 16°
1053. If sin θ = 3.5x and cos θ = 5.5x, find x.
A. 0.1532 B. 0.1534 C. 0.1536 D. 0.1538
1054. Find θ if 2tan θ = ( 1 - tan² θ) cot 56° .
A. 18° B. 16° C. 19° D. 17°
1055. Solve for x if Arctan ( 1 – x ) + Arctan ( 1 + x ) = Arctan ( 1/8 ).
A. 2 B. 4 C. 6 D. 3
1056. If A + B = 180°, then which of the following is true ? sin A = sin B cos A = cos B tan A = tan B
A. (1) only B. (2) only C. (3) only D. all of them 1057. Simplify (sin ½x – cos ½x) ²
A. 1 + sin x B. 1 – sin x C. 1 + cos x D. 1 – cos x 1058. Find the value of Arctan 2cos(Arcsin √3/ 2) .
A. 30° B. 45° C. 60° D. 90°
1059. If sin A = -7/25 where 180° < A < 270°, find tan(A/2).
A. -1/5 B. -5 C. -1/7 D. -7
1060. If sin²x + y = m and cos²x + y = n, find y.
A. (m + n + 1)/2 B. (m + n – 1)/2 C. (m+n)/2 – 1 D. (m+n)/2 +1
1061. Given cos θ = √3/2, find the value of 1 - tan² θ.
A. -2 B. -1/3 C. ½ D. 2/3
1062. What is the value of A between 270° and 360° if 2sin² A – sin A = 1 ?
A. 290° B. 275° C. 300° D. 330°
1063. Evaluate ( sin 0° + sin 1° + sin 2° + … + sin 90°) / ( cos 0° + cos 1° + cos 2° + … + cos 90°)
A. 0 B. 1 C. 2 D. 3
1064. If the supplement of an angle θ is 5/2 of its complement. Find the value of θ.
A. 30° B. 25° C. 20° D. 15°
1065. Express -4 - 4√3 i in trigonometric form.
A. 8 cis 120° B. 8 cis 240° C. 8 cis 150° D. 8 cis 300°
1066. If cos A = -15/17 and A is in quadrant III, find cos ½ A.
A. 0.29054 B. 0.24125 C. 0.24254 D.0.24354
1067. If sin A = 3/5 and cos B = 5/13, find sin (A + B).
A. 0.388 B. 0.865 C. 0.650 D. 0.969
1068. Simplify (sin 2x) / ( 1 + cos 2x)
A. cot x B. tan x C. tan 2x D. 1
1069. A pole which leans to the sun by 10° 15’ from the vertical casts a shadow of 9.43 m on the level ground when the angle of elevation of the sun is 54°50’. The length of the pole is
A. 15.3 m B. 16.3 m C. 17.3 m D. 18.3 m
1070. Triangle ABC has sides a, b and c. If a = 75 m, b = 100 m and the angle opposite side a is 32°, find the angle opposite side c.
A. 93° B. 80° C. 103° D. 100°
1071. If the cosine of angle x is 3/5, then the value of the sine of x/2 is
A. 0.500 B. 0.361 C. 0.215 D. 0.447
1072. If 82° + 0.35x = Arctan( cot 0.45x ), find x.
A. 11° B. 10° C. 12° D. 13°
1073. If sec A = -5/4, A in quadrant II, find tan 2A.
A.24/7 B.25/7 C.-25/7 D.-24/7
1074. Evaluate cos( Arcsin 3/5 + Arctan 8/15 )
A. 34/85 B. 35/85 C. 36/85 D.37/85
1075. If sin x = ¼ , find the value of 4sin(x/2)cos(x/2).
A. 1/8 B. 1/3 C. ½ D. 1/6
1076. If Arcsin( x – 2 ) = π/6, find x.
A. 5/4 B. 5/3 C. 5/2 D. 5/6
1077. The trigonometric expression ( 1 - tan²x ) / ( 1 + tan²x ) is equal to
A. sin1/2x B. sin2x C. cos1/2x D. cos2x
1078. If x + y = 90°, then ( sinx tan y ) / ( sin y tan x ) is equal to
A. tanx B. 1/tanx C. –tanx D. -1/tanx
1079. Twelve round holes are bored through a square piece of steel plate. Their centers are equally spaced on the circumference of a circle 18 cm in diameter. Find the distance between the centers of two consecutive holes.
A. 4.33 cm B. 4.44 cm C. 4.55 cm D. 4.66 cm
1080. Two sides and the included angle of a triangle are measured to be 11 cm, 20 cm and 112°
respectively. Find the length of the third side.
A. 26.19 cm B. 24.14 cm C. 23.16 cm D. 22.15 cm 1081. The rationalized value of ( 4 - 4√3 i ) / ( -2√3 + 2i ) is
A. √3 + i B. -√3 + i C. -√3 – i D. √3 – i
1082. If Arctan(2x) + Arctan(x) = π/4, find x.
A. 0.261 B. 0.271 C. 0.281 D. 0.291
1083. A ladder leans against the wall of a building with its lower end 4 m from the building.
How long is the ladder if it makes an angle of 70° with the ground?
A. 12.3 m B. 13.5 m C. 11.7 m D. 10.8 m
1084. Find the product of (4cis120°)(2cis30°) in rectangular form.
A. -4(√3 + i) B. -4(√3 – i) C. 4(√3 + i) D. 4(√3 – i) 1085. Solve for x if x = (tanθ + cotθ) ² sinθ - tan²θ
A. 4 B. 3 C. 2 D. 1
1086. If ysinx = a and ycosx = b, find y in terms of a and b.
A. a + b B. a² + b² C. √a² + b² D. √a + b
1087. If tan(Arctanx + Arctan ¼) = 7/11, find x.
A. 1/3 B. ¼ C. 1/5 D. ½
1088. if tanθ = √3, θ in quadrant III, find the value of (1 + cosθ) / (1 – cosθ).
A. ½ B. ¼ C. 1/3 D. 1/5
1089. From the top of a lighthouse 37 m above sea level, the angle of depression of a boat is 15°.
How far is the boat from the lighthouse?
A. 138.1 m B. 137.2 m C. 136.3 m D. 135.4 m
1090. The angles B and C of a triangle ABC are 50°30’ and 122°09’ respectively and BC = 9, find the length of AB.
A. 57.36 B. 58.46 C. 59.56 D. 60.66
1091. If the product of csc(x/2) and cos(x/3 + 60°) is equal to 1, find the value of x.
A. 46° B. 36° C. 26° D. 16°
1092. If Arctanx + Arctan(1/3) = 45°, find x.
A. ½ B. ¼ C. 1/5 D. 1/6
1093. If cscθ = 2 and cosθ < 0, then ( secθ + tanθ ) / ( secθ – tanθ ) =
A. 2 B. 3 C. 4 D. 5
1094. Evaluate [6( cos80° + isin80° ) / 3( cos35° + isin35° )]
A. √2 ( 1 + i ) B. √2 ( 1 – i ) C. 2 ( 1 + i ) D. 2 ( 1- i ) 1095. If sin(x + 10°) = cos3x, then x =
A. 23° B. 22° C. 21° D. 20°
1096. If cos(x + y) = 0.17 and cosx = 0.50, find sin y.
A. 0.2355 B. 0.3455 C. 0.4344 D. 0.4233
1097. If sin A + sin B = 1 and sin A – sin B = 1, find A.
A. 60° B. 70° C. 80° D. 90°
1098. At a certain instant, a lighthouse is 4 miles north of a ship which is traveling directly east.
If after 10 minutes, the bearing of the lighthouse is found to be North 21 degree 15 minutes West, find the speed of the ship in miles per hour.
A. 11.3 mph B. 10.3 mph C. 9.3 mph D. 8.3 mph 1099. Simplify ( sec A + csc A ) / ( 1 + tan A )
A. csc A B. sec A C. cot A D. sin A
1100. Evaluate [2(cos60° + isin60°)]³
A. 8 B. -8 C. 8i D. -8i
1101. Evaluate (1 + i)^8
A. -16i B. 16i C. -16 D. 16
1102. Two buildings with flat roofs are 15 m apart. From the edge of the roof of the lower building, the angle of elevation of the edge of the roof of the taller building is 32°. How high is the taller building if the lower building is 18 m high?
A. 26.4 m B. 27.4 m C. 28.4 m D. 29.4 m
1103. If two sides of a triangle are each equal to 8 units and the included angle is 70°, find the third side.
A. 6.15 B. 7.16 C. 8.17 D. 9.18
1104. Express sin(2Arccosx) in terms of x.
A. 2x√1 + x² B. 3x√1 + x² C. 2x√1 - x² D. 3x√1 - x² 1105. Transform Arctanx + Arctany = pi/4 into an algebraic equation
A. x + xy + y = 1 B. x + xy –y = 1 C. x – xy + = 1 D. x – xy-y =1 1106. A tower 28.65 m high is situated on the bank of a river. The angle of depression of an object on the opposite bank of the river is 25°20’. Find the width of the river.
A. 62.50 m B. 60.52 m C. 65.20 m D. 63.25 m
1107. Two cars start at the same time from the same station and move along straight roads that form an angle of 30°, one car at the rate of 30 kph and the other at the rate of 40 kph. How far apart are the cars at the end of half an hour ?
A. 10.17 km B. 10.27 km C. 10.37 km D. 10.47 km 1108. Given: sec2θ = √10 and 2θ in quadrant IV
Find : cos4θ
A. -0.60 B. -0.70 C. -0.80 D. -0.90
1109. The bearing of B from A is N20°E, the bearing of C from B is S30°E and the bearing of A from C is S40°W. If AB = 10, find the area of triangle ABC.
A. 14.95 B. 13.94 C. 12.93 D. 11.92
1110. Two ships start from the same point, one going south and the other North 28° East. If the speed of the first ship is 12 kph and the second ship is 16 kph, find the distance between them after 45 minutes.
A. 17.3 km B. 18.5 km C. 19.2 km D. 20.4 km
1111. If tanθ = ´ and θ is in the 1st quadrant, find tan 4θ.
A. -24/7 B. -20/7 C. -23/7 D.-22/7
1112. Find the height of a tree if the angle of elevation of its top changes from 20° to 40° as the observer advances 23m toward its base.
A. 138.5 m B. 148.5 m C. 158.5 m D. 159.5 m
1113 If 77° + (2x/5) = Arccos(sin x/4) , find x.
A. 21° B. 20° C. 19° D. 18°
1114. Evaluate tan (Arccos(12/13) – Arcsin(4/5))
A. -33/56 B. -33/55 C. -33/54 D. -33/53
1115. Three times the sine of an angle is equal to twice the square of the cosine of the same angle. Find the angle.
A. 20° B. 25° C. 30° D. 35°
1116. Stations A and B are 1000 m apart on a straight road running from eat to west. From A, the bearing of a tower at C is 32° west of north and from B, the bearing of C is 26° north of east.
Find the shortest distance of the tower at C from the road.
A. 243.92 m B. 253.92 m C. 263.92 m D. 273.92 m 1117.If tan35° = y, then (tan145° - tan125°) / (1 + tan145°tan125°) =
A.(1 + y²) / 2y B.(1 - y²) / 2y C.(y²-1) / 2y D. (2y-1)/2y 1118. A tree stands vertically on a hillside which makes an angle of 22° with the horizontal.
From a point 60 ft down the hill directly from the base of the tree, the angle of elevation of the top of the tree is 55°. How high is the tree ?
A. 56.97 ft B. 57.96 ft C. 59.76 ft D. 57.69 ft 1119. If cos 2A = √m , find cos 8A.
A. 8m² + 8m + 1 B. 8m² + 8m – 1 C. 8m² -8m + 1 D. 8m² - 8m -1 1120. The angle of triangle ABC are in the ratio 5:10:21 and the side opposite the smallest angle is 5. Find the side opposite the largest angle.
A. 13.41 B. 14.31 C. 13.14 D. 11.43
1121. On the top of a cliff, the farthest distance that can be seen on the surface of the earth is 60 miles. How high is the cliff if the radius of the earth is taken to be 4000 miles ?
A. 0.41 mi B. 0.43 mi C. 0.45 mi D. 0.47
1122. Two towers are of equal height. At a point P on level ground between them, the angle of elevation of the top of the nearer tower is 60° and at a point M 24 meters directly away from point P, the angle of elevation of the top of the nearer tower is 45°. How high is each tower ?
A. 20.8 m B. 19.8 m C. 18.8 m D. 17.8 m
1123. A quadrilateral ABCD has its side AB perpendicular to side BC at B and its side AD perpendicular to side CD at D. If angle BAD equals 60°, AB = 10 m and AD = 12 m, find the distance (diagonal) from A to C.
A. 11.96 m B. 12.86 m C. 13.76 m D. 14.66 m
1124. The sides of triangle ABC are AB = 5, BC = 12 and AC = 10. Find the length of the line segment drawn from vertex A and bisecting BC.
A. 5.15 B. 5.25 C. 5.35 D.5.45 1125. Express 1/2 (1 - √3 i ) in trigonometric form.
A. cis 120° B. cis 240° C. cis 300° D. cis 315°
1126. If versinθ = x and 1 – sinθ = ´ , find x if θ < 90°.
A. 0.124 B. 0.134 C. 0.154 D. 0.164
1127. Two points A and B, 150 m apart lie on the same side of a tower on a hill and in a
horizontal line passing directly under the tower. The angles of elevation of the top and bottom of the tower viewed from B are 42° and 34° respectively and at A, the angle of elevation of the bottom is 10°. Find the height of the tower.
A. 7.3 m B. 8.3 m C. 9.3 m D. 10.3 m
1128. A point P is at a distance of 4, 5 and 6 from the vertices of an equilateral triangle of side of x. Find x.
A. 8.5 B. 9.5 C. 7.5 D. 10.5
1129. A quadrilateral ABCD has its sides AB and BC perpendicular to each other at B. Side AD makes an angle of 45° with the vertical while side CD makes an angle of 70° with the horizontal.
If AB = 15 and BC = 10, find the length of side CD.
A. 31.5 B. 51.5 C. 61.5 D. 41.5
1130. A clock has a minute hand 16 cm long and an hour hand 11 cm long. Find the distance between the outer tips of the hands at 2:30 o’clock.
A. 19.6 cm B. 20.6 cm C. 21.6 cm D. 22.6 cm
1131. If rcosxsiny = a, rcosxcosy = b and rsinx = c, find r.
A. √a² - b² - c² B. √a² + b² -c² C. √a² - b² + c² D. √a²+b²+c² 1132. From the top of a tower 18 m high, the angles of depression of two objects situated in the horizontal line with the base of the tower and on the same side, are 30 and 45 degrees. Find the distance between the two objects.
A. 13.18 m B. 13.28 m C. 13.38 m D. 13.48 m
1133. The sum of the sines of two angles A and B is 3/2 while the sum of the cosines of the angles is √3 /2 . Find A.
A. 60° B. 30° C. 90° D. 45°
1134. Evaluate tan( Arcsec √5 – Arccot 2 )
A. 3/7 B. 3/5 C. ¾ D. 3/6
1135. What is the greatest distance on the surface of the earth that can be seen from the top of Mayon volcano which is 2.4 kilometers high if the radius of the earth is 6370 km ?
A. 159.7 km B. 174.8 km C. 179.7 km D. 189.7 km
1136. A pole stands on a plane which makes an angle of 15° with the horizontal. A wire from the top of the pole is anchored on a point 8 m from the foot of the pole. If the angle between the wire and the plane is 30 degrees, find the length of the wire.
A. 10.93 m B. 11.93 m C. 12.93 m D. 13.93 m
1137. If sin x + sin y = ½ and cos x – cos y = 1, find x.
A. 15° B. 20° C. 25° D. 30°
1138. A tower standing on level ground is due north of point A and due east of point B. At A and B, the angles of elevation of the top of the tower are 60° and 45° respectively. If AB = 20 , find the height of the tower.
A. 18.32 m B. 17.32 m C. 16.32 m D. 15.32 m
1139. If cot(80° - x/2) cot(2x/3) = 1, find x.
A. 30° B. 60° C. 90° D. 45°
1140. If Arctan z = x/2, find cos x in terms of z.
A. (1 + z²) / (1 - z²) B. (1 - z²) / (1 + z²) C. (z² + 1) / (z² - 1) D. (z² - 1) / (z² + 1)
1141. A flagstaff stands on the top of a house 15 m high. From a point on the plane on which thee house stands., the angles of elevation of the top and bottom of the flagstaff are found to be 60° and 45° respectively. Find the height of the flagstaff.
A. 10.98 m B. 11.87 m C. 12.76 m D. 13.25 m 1142. Two observers 100m apart and facing each other on a horizontal plane, observer at the same time the angles of elevation of a balloon in their vertical to be 58° and 44°. Find the height of the balloon .
A. 60.23 m B. 59.34 m C. 61.31 m D. 58.75 m
1143. From a point outside an equilateral triangle, the distances of the vertices are 10 m, 18 m and 10 m respectively. Find the side of the triangle.
A. 19.94 m B. 20.94 m C. 21.94 m D. 22.94 m
1144. A spherical triangle which contains at least one side equal to a right angle is called A. a right triangle C. an isosceles triangle
B. a polar triangle D. a quadrantal triangle
1145. If A, B and C are the angles of a spherical triangle, then which of the following is true ? A. 180° < A + B + C < 360° C. 0° < A + B + C < 360°
B. 180° < A + B + C < 540° D. 0° < A + B + C < 180°
1146. The angular distance of the horizon from the zenith is equal to how many degrees ?
A. 45° B. 60° C. 90° D. 180°
1147. The point on the celestial sphere directly above the observer is called the
A. zenith B. nadir C. pole D. azimuth
1148. The small circle parallel to the equator is called the
A. equinox B.parallel of latitude C.meridian D.horizon 1149. If a, b and c are the sides of a spherical triangle, then
A. a + b + c < 180° B. a + b + c < 360° C. a + b + c < 540° D. a+b+c< 90°
1150. The point on the celestial sphere diametrically opposite the zenith is called the
A. south pole B. nadir C. azimuth D. north pole
1151. It is the angular distance of a heavenly body from the celestial equator.
A. declination B. altitude C. latitude D. colatitude 1152. At sunset or at sunrise, the astronomical triangle is
A. an isosceles triangle C. a right triangle B. a quadrantal triangle D. an oblique triangle 1153. The azimuth angle is always less than
A. 90° B. 180° C. 360° D. 540°
1154. A great circle which passes through the celestial poles and a heavenly body B is called the ________ of B.
A. vertical circle B. hour circle C. longitude D. horizon 1155. The angular distance of a point on the celestial sphere from the horizon is called its
A. longitude C. latitude
B. altitude D. polar distance
1156. It is the angle at the zenith from the upper branch of the observer's meridian toward the east to the vertical circle of the heavenly body.
A. quadrantal angle C. hour angle
B. polar angle D. azimuth
1157. The zenith distance of a star is the complement of its
A. declination C. altitude
B. polar distance D. latitude
1158. Which of the following given sets of parts of a spherical triangle is possible in order to define the triangle ?
A. A = 55°, B = 65°, C = 60°
B. a = 110°, b = 135°, c = 130°
C. A = 160°, B = 65°, C = 90°
D. a = 120°, b = 150°, c = 60°
1159. The complement of the declination of a star is called the A. polar distance C. longitude B. zenith distance D. altitude
1160. A 90-degree arc on the terrestrial sphere is equal to how many nautical miles ?
A. 3400 B. 4400 C. 5400 D. 6400
1161. How far in statute miles is a place at latitude 40° N from the equator ?
A. 2764.8 B. 2846.7 C. 2684.7 D. 2486.7
1162. Find the distance in nautical miles between A ( 40°30'N, 60°E ) and B (80°20'S, 60°E)
A. 6250 B. 7250 C. 8250 D. 9250
1163. Express 82°26' in nautical miles.
A. 4946 B. 4694 C. 4964 D. 4496
1164. If a place is 12°S of the equator, find its distance in nautical miles from the north pole.
A. 5130 B. 6120 C. 7110 D. 8100
1165. Find the difference in longitude between the following places: M(34°54'33" N, 56°12'51" W)
P(30°20'46" N, 87°18'20" W)
A. 31°05'29" B. 31°06'28" C. 31°07'27" D. 31°08'26"
1166. Find the difference in latitude between the places given in problem 22.
A. 4°32'46" B. 4°33'47" C. 4°31'48" D. 4°30'49"
1167. If an observer is 840 nautical miles south of the equator, find his latitude.
A. 12° S B. 13° S C. 14° S D. 15° S
1168. How far apart are two points on the equator one in longitude 40° East and the other in longitude 150° West ?
A. 190° B. 180° C. 170° D. 160°
1169. Express 3^h 11^m 55^s in angle units.
A. 45°47'58" B. 58°47'45" C. 47°45'58" D. 47°58'45"
1170. Express 260°34' in time units.
A. 17^h 22^m 16^s C. 17^h 26^m 21^s
B. 17^h 16^m 22^s D. 17^h 21^m 26^s
1171. The plane of a small circle on a sphere of radius 25 cm is 7 cm from the center of the sphere. Find the radius of the small circle.
A. 22 cm B. 23 cm C. 24 cm D. 25 cm
1172. Find the area of a spherical triangle ABC on the surface of a sphere of raidus 10 where A = 119°37', B = 38°43' and C = 34°23'.
A. 23.18 B. 22.19 C. 21.16 D. 24.13
1173. An hour-angle of one hour is equal to
A. 14° B. 13° C. 15° D. 16°
1174.The plane of a small circle on a sphere of radius 10 cm is 5 cm from the center of the sphere. Find the area of the small circle.
A. 55π B. 65π C. 75π D. 85π
1175. If the radius of the earth is 3960 miles, find the radius of a parallel of latitude 50° north.
A. 2445.44 mi B. 2554.44 mi C. 2455.44 mi D. 2545.44 mi 1176. Use Napier's rule to find a formula for finding angle B of a right spherical triangle when angle A and side c are given.
A. tan B = cos c tan A C. cot B = cos c tan A B. cot B = sin c tan A D. tan B = sin c cot A
1177. Given a right triangle with angles A = 63°15' and B = 135°34'. Find side b.
A. 134.1° B. 143.1° C. 131.4° D. 141.3°
1178. The two sides of a right spherical triangle are 86°40' and 32°41'. Find the angle opposite the first given side.
A. 88°12' B. 87°11' C. 86°10' D. 85°09'
1179. If the angles of a spherical triangle are A = 74°21' , B = 83°41' and C = 58°39', find side c.
A. 55°54' B. 54°55' C. 45°55' D. 55°45'
1180. The sides of an oblique spherical triangle ABC are given as follows: a = 51°31' , b = 36°47' and c = 80°12'. Find A.
A. 32.35° B. 33.45° C. 34.55° D. 35.56°
1181. Find the distance of Manila(14°36' N, 121°05' E) from Hongkong(22°18' N, 114°10' E) in kilometers.
A. 1123.42 km B.1124.32 km C.1231.24 km D.1321.42km 1182. If a boat sails N 30° W until the departure is 20 miles, what distance does it sail?
A. 55 mi B. 50 mi C. 45 mi D. 40 mi
1183. A ship in latitude 50° N sails 80 nautical miles due East. Find the resulting change in longitude.
A. 2.05° E B. 2.07° E C. 2.09° E D. 2.03° E
1184. Find the longitude of an observer if his local apparent time is 10:36:41 AM and the local Greenwich time is 4:23:12 AM.
A. 93°22'15" E C. 91°22'15" E B. 92°22'15" E D. 90°22'15" E
1185. A ship in latitude 32° N sails due East intil it has made good a difference in longitude of 2°35' . Find the departure.
A. 128.42 nm B. 129.43 nm C. 130.44 nm D. 131.45 nm 1186. Given a spherical triangle ABC with a = 68°27' , b = 87°32' and C = 97°53'. Find c.
A. 96.41° B. 95.14° C. 94.61° D. 93.65°
1187. Find the area of a spherical triangle on the surface of a sphere of radius 10 where a =
1187. Find the area of a spherical triangle on the surface of a sphere of radius 10 where a =