Deck 7: Applications of Trigonometry and Vectors

Question 1

Vector v has the given magnitude and direction. Find the horizontal or vertical component of v, as indicated, if
direction angle of v from the horizontal. Round to the nearest tenth when necessary.

Answer 1

B

Question 2

Solve the problem.
If u = -5, 3 , v = 6, -6 , and w = -3, 12 , evaluate u ·w + v ·w.

A)-39
B)-43
C)-34
D)-32

Answer 2

A

Question 3

Solve the problem.
Two airplanes leave an airport at the same time, one going northwest (bearing 135°) at 414 mph and the other going east at 343 mph. How far apart are the planes after 4 hours (to the nearest
Mile)?

A)2330 mi
B)2800 mi
C)2197 mi
D)700 mi

Answer 3

B

Question 4

Graph the polar equation for in [0°, 360°).
State the formulas for the area of a triangle ABC that were derived using the law of sines.

Question 5

Solve the problem.
A guy wire to a tower makes a 67° angle with level ground. At a point 33 ft farther from the tower than the wire but on the same side of the base as the wire, the angle of elevation to the top of the
Pole is 38°. Find the wire length (to the nearest foot).

A)89 ft
B)84 ft
C)42 ft
D)47 ft

Question 6

Graph the polar equation for in [0°, 360°).
Why can't the Pythagorean theorem be used to solve an oblique triangle?

Question 7

Graph the polar equation for in [0°, 360°).
A trigonometry student makes the statement, "If we know any two sides and one angle of a
triangle, then the triangle is uniquely determined." Is this a valid statement? Explain,
referring to the congruence axioms.

Question 8

Solve the problem.
Suppose you would like to cross a 219-foot wide river in a boat. Assume that the boat can travel 38 mph relative to the water and that the current is flowing west at the rate of 7 mph. What bearing
Should be chosen so that the boat will land at a point exactly across from its starting point? Give
Your answer to the nearest tenth of a degree.

A)10.6°
B)7.1°
C)8.5°
D)79.4°

Question 9

Determine the number of triangles ABC possible with the given parts.
b = 42, c = 52, B = 106°

A)0
B)1
C)3
D)2

Question 10

Solve the problem.
A painter needs to cover a triangular region 61 meters by 69 meters by 72 meters. A can of paint covers 70 square meters. How many cans will be needed?

A)14 cans
B)318 cans
C)3 cans
D)28 cans

Question 11

Draw a sketch to represent the vector. Refer to the vectors pictured here.
3d

Question 12

Find the area of triangle ABC with the given parts. Round to the nearest tenth when necessary.
a = 13.6 cm b = 18.4 cm
C = 15.3 cm

Question 13

Solve the problem.
Starting at point A, a ship sails 48 km on a bearing of 217°, then turns and sails 15 km on a bearing of 280°. Find the distance of the ship from point A.

A)63 km
B)71 km
C)56 km
D)8 km

Question 14

Determine whether there is sufficient information for solving a triangle, with the given combination of angles and sides,
by the law of sines.
b, C, and B

A)Yes
B)No

Question 15

Solve the problem.
A plane is heading due south with an airspeed of 245 mph. A wind from a direction of 53° is blowing at 13 mph. Find the bearing of the plane.

A)177°
B)182°
C)88°
D)93°

Question 16

Write the vector in the form . If necessary, round values to the nearest hundredth.

Question 17

Find the missing parts of the triangle.
B = 44°20' b = 12.30
A = 31.26
If necessary, round side lengths to the nearest hundredth.

A)A = 43°10', C = 94°30', c = 45.06
B)A = 40°10', C = 95°30', c = 40.56
C)no such triangle
D)A = 42°10', C = 93°30', c = 43.56

Question 18

Solve the problem.
It is 4.7 km from Lighthouse A to Port B. The bearing of the port from the lighthouse is N73°E. A ship has sailed due west from the port and its bearing from the lighthouse is N31°E. How far has
The ship sailed from the port? Round to the nearest tenth of a kilometer.

A)3.1 km
B)3.5 km
C)2.7 km
D)3.7 km

Question 19

Find the missing parts of the triangle.
B = 29.5° b = 19.45
A = 19.75
If necessary, round angles to the nearest tenth and side lengths to the nearest hundredth.

A)no such triangle
B)A = 30°, C = 120.5°, c = 34.03
C)A1 = 30°, C1 = 120.5°, c1 = 34.03;
D)A = 150°, C = 0.5°, c = 0.34 A2 = 150°, C2 = 0.5°, c2 = 0.34

Question 20

Find the angle between the pair of vectors to the nearest tenth of a degree.
6i + 3j, -2i + 3j

A)104.4°
B)98.1°
C)97.1°
D)150.3°

Question 21

Find the dot product for the pair of vectors.

Question 22

Solve the problem.
A force of 627 lb is required to pull a boat up a ramp inclined at 23° with the horizontal. How much does the boat weigh?

A)604 lb
B)577 lb
C)2182 lb
D)1605 lb

Question 23

Solve the problem.
Two people are carrying a box. One person exerts a force of 129 pounds at an angle of 49.4° with the horizontal. The other person exerts a force of 98 pounds at an angle of 52.4°. Find the weight of
The box.

A)266 lb
B)227 lb
C)104 lb
D)177 lb

Question 24

Solve the problem.

Question 25

Find the magnitude and direction angle (to the nearest tenth) for each vector. Give the measure of the direction angle as
an angle in [0,360°].

Question 26

Find the missing parts of the triangle.
A = 20° a = 35 km
B = 50 km
If necessary, round angles to the nearest whole number and side lengths to the nearest km.

Question 27

Find the missing parts of the triangle.

Question 28

Find the magnitude and direction angle (to the nearest tenth) for each vector. Give the measure of the direction angle as
an angle in [0,360°].
-16, 0

A)16; 180°
B)256; 180°
C)16; 270°
D)16; 90°

Question 29

Solve the problem.
A room in the shape of a triangle has sides of length 14.5 yards, 13.8 yards, and 18.9 yards. If carpeting costs $18.85 per square yard, padding costs $2.40 per square yard, and there is no charge
For installation, how much, to the nearest dollar, will it cost to carpet the room?

A)$99
B)$435
C)$3782
D)$2114

Question 30

Write the vector in the form . If necessary, round values to the nearest hundredth.

Question 31

Find the component form of the indicated vector.

Question 32

Determine whether there is sufficient information for solving a triangle, with the given combination of angles and sides,
by the law of sines.
B, a, and c

A)Yes
B)No

Question 33

Find the missing parts of the triangle.
A = 78° a = 36 yd
B = 73 yd
If necessary, round angles to the nearest degree and side lengths to the nearest yard.

A)B = 29°, C = 73°, c = 81 yd
B)B = 29°, C = 73°, c = 75 yd
C)B = 73°, C = 29°, c = 79 yd
D)no such triangle

Question 34

Determine whether there is sufficient information for solving a triangle, with the given combination of angles and sides,
by the law of sines.
b, B, and c

A)Yes
B)No

Question 35

Determine the number of triangles ABC possible with the given parts.

Question 36

Write the vector in the form ai + bj.
-5, 6

A)5i + 6j
B)6i + 5j
C)-5i + 6j
D)6i - 5j

Question 37

Solve the problem.
To find the distance AB across a river, a distance BC = 346 m is laid off on one side of the river. It is found that B = 115.1° and C = 13.6°. Find AB rounded to the nearest meter.

A)107 m
B)80 m
C)104 m
D)83 m

Question 38

Draw a sketch to represent the vector. Refer to the vectors pictured here.
b + c

Question 39

Use the figure to find the specified vector.
Find a + b.

Question 40

Sketch the vectors u and w with angle between them and sketch the resultant.

Question 41

Solve the problem.

Question 42

Determine the number of triangles ABC possible with the given parts.

Question 43

Find the angle between the pair of vectors to the nearest tenth of a degree.

Question 44

Find the angle between the pair of vectors to the nearest tenth of a degree.

Question 45

Find the indicated vector.

Question 46

Find the indicated vector.

Question 47

Determine whether there is sufficient information for solving a triangle, with the given combination of angles and sides,
by the law of sines.
A, c, and a

A)Yes
B)No

Question 48

Find the missing parts of the triangle.
A = 97.3° b = 15.2 ft
A = 30.7 ft
If necessary, round angles and side lengths to the nearest tenth.

A)B = 29.4°, C = 53.3°, c = 24.8 ft
B)B = 34.4°, C = 48.3°, c = 24.8 ft
C)B = 53.3°, C = 29.4°, c = 24.8 ft
D)no such triangle

Question 49

Find the missing parts of the triangle.

Question 50

Solve the problem.
A hot-air balloon is rising vertically 10 ft/sec while the wind is blowing horizontally at 4 ft/sec. Find the angle that the balloon makes with the horizontal.

A)68.2°
B)34.3°
C)21.8°
D)52.8°

Question 51

Solve the problem.
Two ships leave a harbor together traveling on courses that have an angle of 129° between them. If they each travel 505 miles, how far apart are they (to the nearest mile)?

A)41 mi
B)1824 mi
C)912 mi
D)435 mi

Question 52

Solve the problem.
A solar reflector is made using 29 identical triangular-shaped mirrors, each having sides 1.7 meters, 1.9 meters, 1.1 meters. What is the total surface area of the reflector (to the nearest square
Meter)?

Question 53

Find the indicated vector.

Question 54

Draw a sketch to represent the vector. Refer to the vectors pictured here.

Question 55

Find the magnitude and direction angle (to the nearest tenth) for each vector. Give the measure of the direction angle as
an angle in [0,360°].

Question 56

Find the indicated vector.

Question 57

Determine whether there is sufficient information for solving a triangle, with the given combination of angles and sides,
by the law of sines.

Question 58

Determine whether there is sufficient information for solving a triangle, with the given combination of angles and sides,
by the law of sines.
C, c, and A

A)Yes
B)No

Question 59

Find the magnitude and direction angle (to the nearest tenth) for each vector. Give the measure of the direction angle as
an angle in [0,360°].

Question 60

Determine whether there is sufficient information for solving a triangle, with the given combination of angles and sides,
by the law of sines.
a, b, and c

A)Yes
B)No

Question 61

Find the component form of the indicated vector.

Question 62

Solve the problem.
One rope pulls a barge directly east with a force of 42 newtons, and another rope pulls the barge directly north with a force of 68 newtons. Find the magnitude of the resultant force acting on the
Barge.

A)110 newtons
B)80 newtons
C)2900 newtons
D)26 newtons

Question 63

Write the vector in the form . If necessary, round values to the nearest hundredth.

Question 64

Determine whether there is sufficient information for solving a triangle, with the given combination of angles and sides,
by the law of sines.
b, A, and B

A)Yes
B)No

Question 65

Solve the problem.
Two points A and B are on opposite sides of a building. A surveyor chooses a third point C 77 yd from B and 95 yd from A, with angle ACB measuring 54.3°. How far apart are A and B (to the
Nearest yard)?

A)98 yd
B)107 yd
C)80 yd
D)89 yd

Question 66

Draw a sketch to represent the vector. Refer to the vectors pictured here.
a + d

Question 67

Write the vector in the form . If necessary, round values to the nearest hundredth.

Question 68

Find the dot product for the pair of vectors.

Question 69

Solve the problem.
Points A and B are on opposite sides of a lake. Point C is 80.7 meters from A. The measure of angle BAC is 72°20', and the measure of angle ACB is determined to be 35°50'. Find the distance between
Points A and B (to the nearest meter).

A)23 m
B)25 m
C)48 m
D)50 m

Question 70

Solve the problem.

Question 71

Find the dot product for the pair of vectors.

Question 72

Use the figure to find the specified vector.
Find -a.

Question 73

Find the indicated vector.

Question 74

Find the magnitude and direction angle (to the nearest tenth) for each vector. Give the measure of the direction angle as
an angle in [0,360°].

Question 75

Determine whether the pair of vectors is orthogonal.

Question 76

Solve the problem.
An airplane is sighted at the same time by two ground observers who are 2 miles apart and both directly west of the airplane. They report the angles of elevation as 12° and 21°. How high is the
Airplane? Round to the nearest hundredth of a mile.

A)0.42 mi
B)1.64 mi
C)0.72 mi
D)0.95 mi

Question 77

Draw a sketch to represent the vector. Refer to the vectors pictured here.
a + (c + d)

Question 78

Sketch the vectors u and w with angle between them and sketch the resultant.

Question 79

Find the missing parts of the triangle. Round to the nearest tenth when necessary or to the nearest minute as appropriate.
AB = 321 yd AC = 188 yd
BC = 161 yd

A)No triangle satisfies the given conditions.
B)A = 25.1°, B = 21.3°, C = 133.6°
C)A = 21.3°, B = 12.6°, C = 146.1°
D)A = 21.3°, B = 25.1°, C = 133.6°

Question 80

Find the area of triangle ABC with the given parts. Round to the nearest tenth when necessary.
a = 46 ft b = 54 ft
C = 62 ft