Download G - FundamentalMath

Regents Exam Questions A.A.43: Using Trigonometry to Find an Angle 1
Page 1
www.jmap.org
Name: __________________________________
A.A.42: Trigonometric Ratios: Find the sine, cosine, and tangent ratios of an angle
of a right triangle, given the lengths of the sides
1 The diagram below shows right triangle UPC.
Which ratio represents the sine of
1)
2)
3)
?
4)
2 Which ratio represents
in the right triangle shown below?
1)
2)
3)
4)
Regents Exam Questions A.A.43: Using Trigonometry to Find an Angle 1
Page 2
www.jmap.org
Name: __________________________________
3 Which ratio represents
in the accompanying diagram of
1)
2)
3)
4 In the accompanying diagram of right triangle ABC,
.
What is
?
1)
3)
2)
4)
5 The diagram below shows right triangle ABC.
Which ratio represents the tangent of
1)
2)
3)
4)
?
?
4)
,
,
, and
Regents Exam Questions A.A.43: Using Trigonometry to Find an Angle 1
Page 3
www.jmap.org
Name: __________________________________
6 The diagram below shows right triangle LMP.
Which ratio represents the tangent of
1)
2)
3)
?
4)
7 Right triangle ABC has legs of 8 and 15 and a hypotenuse of 17, as shown in the diagram
below.
The value of the tangent of
is
1) 0.4706 2) 0.5333 3) 0.8824 4) 1.8750
8 In triangle MCT, the measure of
Which ratio represents the sine of
1)
2)
3)
2)
3)
,
, and
.
4)
9 In
, the measure of
the tangent of
?
1)
,
?
4)
,
, and
. Which ratio represents
Regents Exam Questions A.A.43: Using Trigonometry to Find an Angle 1
Page 4
www.jmap.org
Name: __________________________________
10 Which equation shows a correct trigonometric ratio for angle A in the right triangle
below?
1)
11 In
1)
,
2)
. If
2)
1 ANS: 2
REF: 010919ia
2 ANS: 1
REF: 011008ia
3 ANS: 1
REF: 080414a
4 ANS: 1
REF: 010316a
5 ANS: 2
REF: 081112ia
3)
and
3)
4)
, which statement is not true?
4)
Regents Exam Questions A.A.43: Using Trigonometry to Find an Angle 1
Page 5
www.jmap.org
Name: __________________________________
6 ANS: 3
REF: 011226ia
7 ANS: 2
REF: 081026ia
8 ANS: 1
REF: fall0721ia
9 ANS: 2
REF: 061009ia
10 ANS: 3
REF: 011008ia
11 ANS: 4
If
, then
REF: 061224ia
is the hypotenuse, and the triangle is a 3-4-5 triangle.
A.A.44: Using Trigonometry to Find a Side: Find the measure of a side of a right
triangle, given an acute angle and the length of another side
1 In the right triangle shown in the diagram below, what is the value of x to the nearest whole
number?
1)
2)
3)
4)
12
14
21
28
Regents Exam Questions A.A.43: Using Trigonometry to Find an Angle 1
Page 6
www.jmap.org
Name: __________________________________
2 A tree casts a 25-foot shadow on a sunny day, as shown in the diagram below.
If the angle of elevation from the tip of the shadow to the top of the tree is 32°, what is the height
of the tree to the nearest tenth of a foot?
1) 13.2
2) 15.6
3) 21.2
4) 40.0
3 An 8-foot rope is tied from the top of a pole to a stake in the ground, as shown in the diagram
below.
If the rope forms a 57° angle with the ground, what is the height of the pole, to the nearest tenth of
a foot?
1) 4.4
2) 6.7
3) 9.5
4) 12.3
4 A right triangle contains a 38° angle whose adjacent side measures 10 centimeters. What is the
length of the hypotenuse, to the nearest hundredth of a centimeter?
1) 7.88
2) 12.69
3) 12.80
4) 16.24
Regents Exam Questions A.A.43: Using Trigonometry to Find an Angle 1
Page 7
www.jmap.org
Name: __________________________________
5 As shown in the diagram below, a ladder 5 feet long leans against a wall and makes an angle of
65° with the ground. Find, to the nearest tenth of a foot, the distance from the wall to the base of
the ladder.
6 A stake is to be driven into the ground away from the base of a 50-foot pole, as shown in the
diagram below. A wire from the stake on the ground to the top of the pole is to be installed at an
angle of elevation of 52°.
How far away from the base of the pole should the stake be driven in, to the nearest foot? What
will be the length of the wire from the stake to the top of the pole, to the nearest foot?
Regents Exam Questions A.A.43: Using Trigonometry to Find an Angle 1
Page 8
www.jmap.org
Name: __________________________________
7 A hot-air balloon is tied to the ground with two taut (straight) ropes, as shown in the diagram
below. One rope is directly under the balloon and makes a right angle with the ground. The other
rope forms an angle of 50º with the ground.
Determine the height, to the nearest foot, of the balloon directly above the ground. Determine the
distance, to the nearest foot, on the ground between the two ropes.
1
ANS: 3
REF: 010912ia
2 ANS: 2
REF: 080914ia
3 ANS: 2
REF: 061108ia
4 ANS: 2
Regents Exam Questions A.A.43: Using Trigonometry to Find an Angle 1
Page 9
www.jmap.org
Name: __________________________________
REF: 081126ia
5 ANS:
2.1.
REF: 011133ia
6 ANS:
39, 63.
.
REF: 060937ia
7 ANS:
84, 71
REF: 081039ia
1 In the accompanying diagram of right triangle ABC,
and
Which single function could be used to find AB?
1)
2)
3)
4)
2 In right triangle ABC,
. Which equation is true for this triangle?
1)
2)
3)
4)
.
Regents Exam Questions A.A.43: Using Trigonometry to Find an Angle 1
Page 10
www.jmap.org
Name: __________________________________
3 The accompanying diagram shows a ramp 30 feet long leaning against a wall at a construction site.
If the ramp forms an angle of 32° with the ground, how high above the ground, to the nearest
tenth, is the top of the ramp?
1) 15.9 ft
2) 18.7 ft
3) 25.4 ft
4) 56.6 ft
4 The tailgate of a truck is 2 feet above the ground. The incline of a ramp used for loading the
truck is 11°, as shown below. Find, to the nearest tenth of a foot, the length of the ramp.
5 Joe is holding his kite string 3 feet above the ground, as shown in the accompanying diagram.
The distance between his hand and a point directly under the kite is 95 feet. If the angle of
elevation to the kite is 50°, find the height, h, of his kite, to the nearest foot.
Regents Exam Questions A.A.43: Using Trigonometry to Find an Angle 1
Page 11
www.jmap.org
Name: __________________________________
6 A surveyor needs to determine the distance across the pond shown in the accompanying diagram.
She determines that the distance from her position to point P on the south shore of the pond is 175
meters and the angle from her position to point X on the north shore is 32°. Determine the
distance, PX, across the pond, rounded to the nearest meter.
7 Find, to the nearest tenth of a foot, the height of the tree represented in the accompanying diagram.
8 In the accompanying diagram, x represents the length of a ladder that is leaning against a wall of a
building, and y represents the distance from the foot of the ladder to the base of the wall. The
ladder makes a 60° angle with the ground and reaches a point on the wall 17 feet above the
ground. Find the number of feet in x and y.
Regents Exam Questions A.A.43: Using Trigonometry to Find an Angle 1
Page 12
www.jmap.org
Name: __________________________________
9 In the accompanying diagram, a ladder leaning against a building makes an angle of 58º with level
ground. If the distance from the foot of the ladder to the building is 6 feet, find, to the nearest
foot, how far up the building the ladder will reach.
10 As shown in the accompanying diagram, a ladder is leaning against a vertical wall, making an
angle of 70° with the ground and reaching a height of 10.39 feet on the wall. Find, to the nearest
foot, the length of the ladder. Find, to the nearest foot, the distance from the base of the ladder to
the wall.
11 From a point on level ground 25 feet from the base of a tower, the angle of elevation to the top of
the tower is 78°, as shown in the accompanying diagram. Find the height of the tower, to the
nearest tenth of a foot.
Regents Exam Questions A.A.43: Using Trigonometry to Find an Angle 1
Page 13
www.jmap.org
Name: __________________________________
12 A lighthouse is built on the edge of a cliff near the ocean, as shown in the accompanying diagram.
From a boat located 200 feet from the base of the cliff, the angle of elevation to the top of the cliff
is 18° and the angle of elevation to the top of the lighthouse is 28°. What is the height of the
lighthouse, x, to the nearest tenth of a foot?
1
ANS: 1
2 ANS: 4
3 ANS: 1
PTS: 2
4 ANS:
PTS:
2
REF:
PTS: 2
010926a
REF: 018933siii
REF: 080724a
10.5.
PTS: 2
5 ANS:
REF: spring9825a
116.
PTS: 4
6 ANS:
REF: 069934a
109.
PTS: 3
7 ANS:
REF: 060030a
Regents Exam Questions A.A.43: Using Trigonometry to Find an Angle 1
Page 14
www.jmap.org
Name: __________________________________
28.2.
PTS: 4
8 ANS:
REF: 010135a
.
PTS: 4
9 ANS:
REF: 080231a
10.
PTS: 2
REF: 010531a
10 ANS:
Length of ladder = 11 and distance from the base of the ladder to the wall = 4.
.
PTS: 4
11 ANS:
REF: 010638a
117.6.
PTS: 2
12 ANS:
REF: 010735a
41.4.
PTS: 4
REF: 010837a
Regents Exam Questions A.A.43: Using Trigonometry to Find an Angle 1
Page 15
www.jmap.org
Name: __________________________________
1 The angle of elevation from a point 25 feet from the base of a tree on level ground to the top of the
tree is 30°. Which equation can be used to find the height of the tree?
1)
2)
3)
4)
2 In right triangle ABC,
,
, and
. What is the length of the hypotenuse?
1)
2)
3) 8
4)
3 A 10-foot ladder is to be placed against the side of a building. The base of the ladder must be
placed at an angle of 72° with the level ground for a secure footing. Find, to the nearest inch, how
far the base of the ladder should be from the side of the building and how far up the side of the
building the ladder will reach.
4 Draw and label a diagram of the path of an airplane climbing at an angle of 11° with the ground.
Find, to the nearest foot, the ground distance the airplane has traveled when it has attained an
altitude of 400 feet.
5 A tree casts a shadow that is 20 feet long. The angle of elevation from the end of the shadow to
the top of the tree is 66°. Determine the height of the tree, to the nearest foot.
Regents Exam Questions A.A.43: Using Trigonometry to Find an Angle 1
Page 16
www.jmap.org
Name: __________________________________
6 A person measures the angle of depression from the top of a wall to a point on the ground. The
point is located on level ground 62 feet from the base of the wall and the angle of depression is
52°. How high is the wall, to the nearest tenth of a foot?
7 A ship on the ocean surface detects a sunken ship on the ocean floor at an angle of depression of
50°. The distance between the ship on the surface and the sunken ship on the ocean floor is 200
meters. If the ocean floor is level in this area, how far above the ocean floor, to the nearest meter,
is the ship on the surface?
8 At Mogul’s Ski Resort, the beginner’s slope is inclined at an angle of 12.3°, while the advanced
slope is inclined at an angle of 26.4°. If Rudy skis 1,000 meters down the advanced slope while
Valerie skis the same distance on the beginner’s slope, how much longer was the horizontal
distance that Valerie covered?
1) 81.3 m
2) 231.6 m
3) 895.7 m
4) 977.0 m
1 ANS: 1
2 ANS: 3
3 ANS:
114” and 37”.
REF: 080033a
4 ANS:
2,058.
REF: 060419a
REF: 088725siii
.
Regents Exam Questions A.A.43: Using Trigonometry to Find an Angle 1
Page 17
www.jmap.org
Name: __________________________________
REF: 010235a
5 ANS:
45.
REF: 080536a
6 ANS:
79.4.
REF: 060639a
7 ANS:
153.
REF: 080133a
8 ANS: 1
REF: 080108b
Regents Exam Questions A.A.43: Using Trigonometry to Find an Angle 1
Page 18
www.jmap.org
Name: __________________________________
A.A.43: Using Trigonometry to Find an Angle: Determine the measure of an angle
of a right triangle, given the length of any two sides of the triangle
1 Which equation could be used to find the measure of one acute angle in the right triangle
shown below?
1)
2)
3)
4)
2 In the diagram of
shown below,
and
.
To the nearest tenth of a degree, what is the measure of the largest acute angle in the
triangle?
1) 32.0
2) 38.7
3) 51.3
4) 90.0
3 In right triangle ABC shown below,
What is the measure of
1) 31.5
2) 37.7
3) 52.3
4) 58.5
and
.
, to the nearest tenth of a degree?
Regents Exam Questions A.A.43: Using Trigonometry to Find an Angle 1
Page 19
www.jmap.org
Name: __________________________________
4
The center pole of a tent is 8 feet long, and a side of the tent is 12 feet long as
shown in the diagram below.
If a right angle is formed where the center pole meets the ground, what is the measure of
angle A to the nearest degree?
1) 34
2) 42
3) 48
4) 56
5 A communications company is building a 30-foot antenna to carry cell phone
transmissions. As shown in the diagram below, a 50-foot wire from the top of the antenna
to the ground is used to stabilize the antenna.
Find, to the nearest degree, the measure of the angle that the wire makes with the ground.
6 In right triangle ABC,
degree, the measure of
,
,
, and
. Find, to the nearest
.
7 A 28-foot ladder is leaning against a house. The bottom of the ladder is 6 feet from the
base of the house. Find the measure of the angle formed by the ladder and the ground, to
the nearest degree.
Regents Exam Questions A.A.43: Using Trigonometry to Find an Angle 1
Page 20
www.jmap.org
Name: __________________________________
8 A trapezoid is shown below.
Calculate the measure of angle x, to the nearest tenth of a degree.
9 In right triangle ABC shown below,
inches,
inches, and
Find the number of degrees in the measure of angle BAC, to the nearest degree.
Find the length of
to the nearest inch.
.
Regents Exam Questions A.A.42: Trigonometric Ratios 1
www.jmap.org
1 ANS: 1
2 ANS: 3
REF: 080824ia
. A 90º angle is not acute.
REF: 080829ia
3 ANS: 1
4 ANS: 2
REF: 061114ia
REF: 060816ia
5 ANS:
REF: 061033ia
6 ANS:
53.
REF: 011032ia
7 ANS:
78.
REF: 061235ia
8 ANS:
41.8.
REF: 081135ia
9 ANS:
54, 23.
.
REF: 081238ia
A.A.43: Using Trigonometry to Find an Angle: Determine the measure of an angle of a right
triangle, given the length of any two sides of the triangle
Regents Exam Questions A.A.42: Trigonometric Ratios 1
www.jmap.org
1 Cassandra is calculating the measure of angle A in right triangle ABC, as shown in the accompanying diagram.
She knows the lengths of
and
,
If she finds the measure of angle A by solving only one equation, which concept will be used in her calculations?
1) Pythagorean theorem
2)
3)
4)
2 Ron and Francine are building a ramp for performing skateboard stunts, as shown in the accompanying diagram.
The ramp is 7 feet long and 3 feet high. What is the measure of the angle, x, that the ramp makes with the ground,
to the nearest tenth of a degree?
3 A person standing on level ground is 2,000 feet away from the foot of a 420-foot-tall building, as shown in the
accompanying diagram. To the nearest degree, what is the value of x?
4 If a tree 28 meters tall casts a shadow 32 meters long, what is the angle of elevation of the Sun to the nearest
degree?
1) 29
2) 41
3) 50
4) 61
Regents Exam Questions A.A.42: Trigonometric Ratios 1
www.jmap.org
5 As seen in the accompanying diagram, a person can travel from New York City to Buffalo by going north 170
miles to Albany and then west 280 miles to Buffalo.
If an engineer wants to design a highway to connect New York City directly to Buffalo, at what angle, x, would she
need to build the highway? Find the angle to the nearest degree. To the nearest mile, how many miles would be
saved by traveling directly from New York City to Buffalo rather than by traveling first to Albany and then to
Buffalo?
6 In the accompanying diagram, the base of a 15-foot ladder rests on the ground 4 feet from a 6-foot fence.
a If the ladder touches the top of the fence and the side of a building, what angle, to the nearest degree, does the
ladder make with the ground?
b Using the angle found in part a, determine how far the top of the ladder reaches up the side of the building, to the
nearest foot.
7 The accompanying diagram shows a flagpole that stands on level ground. Two cables, r and s, are attached to the
pole at a point 16 feet above the ground. The combined length of the two cables is 50 feet. If cable r is attached to
the ground 12 feet from the base of the pole, what is the measure of the angle, x, to the nearest degree, that cable s
makes with the ground?
Regents Exam Questions A.A.42: Trigonometric Ratios 1
www.jmap.org
1 ANS: 4
2 ANS:
PTS:
2
REF:
060820a
25.4.
PTS: 2
3 ANS:
REF: 060735a
12.
PTS: 3
4 ANS: 2
5 ANS:
REF: 089927a
PTS: 2
59, 122.
.
REF: 068533siii
. The trip from New York City to Buffalo via Albany is 450 (280 +
170) miles. Therefore traveling directly to Buffalo would save (450 – 328) 122 miles.
PTS: 4
6 ANS:
56, 12.
PTS: 4
7 ANS:
32.
PTS: 4
REF: 060231a
.
REF: 010438a
. If the combined length of the two cables is 50 feet, then s is 30 (50 – 20) feet.
REF: 060539a