Deck 7: Applications of Trigonometry

Question 1

Use the following to answer questions :
A radar ship is 25.0 miles off shore from a major port when a large fleet of ships leaves the port at the 40.0° angle shown.
If the maximum range of the ship's radar is 16.0 miles, will the departing fleet be detected?

A) yes
B) no

Answer 1

no

Question 2

Use the following to answer questions :
In $\angle$ A = 60°, and side c = 26 ft.

-How many triangles can be formed if side a = 10 ft?

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

Answer 2

0

Question 3

Use the following to answer questions :
In $\angle$ A = 60°, and side c = 26 ft.

-How many triangles can be formed if side a = 23 ft?

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

Answer 3

1

Question 4

Solve using the law of sines and a scaled drawing. Round to the nearest tenth. If two triangles exist, solve both completely.
side c = 27.5 mi
$\angle$ A = 44°
side a = 10.1 mi

Question 5

Solve using the law of sines and a scaled drawing. If two triangles exist, solve both completely.
side a = 23.6 yd
$\angle$ A = 30°
side c = 47.2 yd

Question 6

Solve the triangle using the law of sines. If the law of sines cannot be used, state why. Round sides to the nearest tenth.

Question 7

Assume the law of sines is being applied to solve a triangle. Solve for A (if possible), then determine if a second angle (0° < $\theta$ < 180°) exists that also satisfies the proportion. Round to the nearest tenth of a degree.

A) 41.1°
B) 41.1°, 138.9°
C) 41.7°, 138.3°
D) not possible

Question 8

Determine the length to the nearest tenth of a foot of both rafters in the diagram. 42 feet

Question 9

Solve the following equation for a. Round to the nearest hundredth.

A) a $\approx$ 6.01
B) a $\approx$ 6.96
C) a $\approx$ 7.67
D) a $\approx$ 8.09

Question 10

Use the following to answer questions :
In $\angle$ A = 60°, and side c = 26 ft.

-How many triangles can be formed if side a = 17 ft?

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

Question 11

Use the following to answer questions :
A radar ship is 25.0 miles off shore from a major port when a large fleet of ships leaves the port at the 40.0° angle shown.

-If the maximum range of the ship's radar is 18 miles, approximately how far from port (to the nearest tenth of a mile) is the fleet when it is first detected?

A). $\approx$ 11.0 mi
B). $\approx$ 14.6 mi
C). $\approx$ 23.9 mi
D). $\approx$ 27.3 mi

Question 12

Determine whether the law of cosines can be used to begin the solution process for the triangle.

A) yes
B) no

Question 13

Assume the law of sines is being applied to solve a triangle. Solve for B (if possible), then determine if a second angle (0° < $\theta$ < 180°) exists that also satisfies the proportion. Round to the nearest tenth of a degree.

Question 14

Use the following to answer questions :
In $\angle$ A = 60°, and side c = 26 ft.

-Assuming that side c is the longest side, what length for side a will produce a right triangle?

A) 11 ft
B) ft
C) ft
D) 44 ft

Question 15

Determine whether the law of cosines can be used to begin the solution process for the triangle.

A) yes
B) no

Question 16

Solve the triangle using the law of sines. If the law of sines cannot be used, state why. Round sides to the nearest tenth.
$\angle$ B = 21°
side a = 18 yd
$\angle$ C = 84°

Question 17

Solve using the law of sines and a scaled drawing. Round to the nearest tenth. If two triangles exist, solve both completely.
side c = 25.0 ft
$\angle$ C = 62°
side b = 26.3 ft

Question 18

Solve using the law of sines and a scaled drawing. Round to the nearest tenth. If two triangles exist, solve both completely. side b = 20.0 mi
$\angle$ B = 51°
Side a = 21.6 mi

A) A $\approx$ 57.1°, C $\approx$ 71.9°
C $\approx$ 24.5 mi
B) A $\approx$ 122.9° C $\approx$ 6.1°
C $\approx$ 2.7 mi
C) A $\approx$ 57.1° C $\approx$ 71.9°
C $\approx$ 24.5 mi
Or
A $\approx$ 122.9°
C $\approx$ 6.1°
C $\approx$ 2.7 mi
D) not possible

Question 19

Solve for B (0 < B < 90°), if possible. Round to the nearest tenth of a degree.

A) 53.6°
B) 54.4°
C) 55.1°
D) not possible

Question 20

Solve the triangle using the law of sines. Round sides to the nearest tenth. side a = 5 m
$\angle$ A = 56°
$\angle$ B = 41°

A). $\angle$ C = 83° b $\approx$ 3.3 m
C $\approx$ 5.4 m
B). $\angle$ C = 83° b $\approx$ 3.8 m
C $\approx$ 6.4 m
C). $\angle$ C = 83° b $\approx$ 4.0 m
C $\approx$ 6.0 m
D). $\angle$ C = 83° b $\approx$ 4.5 m
C $\approx$ 5.8 m

Question 21

The magnitude of vector v is given, along with the quadrant of the terminal point and the angle it makes with the nearest x-axis. Find the horizontal and vertical components of v and write the results in component form. Round to one decimal place. |v| = 17; $\theta$ = 78°; QIII

A) v =
B) v =
C) v =
D) v =

Question 22

Solve the triangle using the law of cosines. Round to tenths.

A) A $\approx$ 22.0° C $\approx$ 138.0°
B $\approx$ 80.7 cm
B) A $\approx$ 158.0° C $\approx$ 2.0°
B $\approx$ 80.7 cm
C) A $\approx$ 56.0° C $\approx$ 104.0°
B $\approx$ 36.4 cm
D) A $\approx$ 124.0° C $\approx$ 36.0°
B $\approx$ 36.4 cm

Question 23

Use the following to answer questions :
u = ; v =
Compute u - v.

A)
B)
C)
D)

Question 24

Use the following to answer questions :
v =
Graph the vector.

Question 25

Determine whether the law of cosines can be used to begin the solution process for the triangle.

A) yes
B) no

Question 26

Two tractors are pulling at a stump in an effort to clear land for more crops. The Massey-Ferguson is pulling with a force of 200 N, while the John Deere is pulling with a force of 250 N. The chains attached to the stump and each tractor form a 28° angle. Represent this situation using geometric vectors.

Question 27

u = ; v = (a) Compute u + v.
(b) Illustrate u + v graphically.

Question 28

Use the following to answer questions :
u = ; v =
Compute u + v.

A)
B)
C)
D)

Question 29

Use the following to answer questions :
Vector v = has initial point (2, 5).
Find the coordinates of the terminal point of the vector.

A) (-7, 9)
B) (1, 7)
C) (-1, -7)
D) (-7, 7)

Question 30

Determine whether the law of cosines can be used to begin the solution process for the triangle.

A) yes
B) no

Question 31

Two planes leave an airport at the same time. One travels due west (bearing 270°) with a cruising speed of 420 mph. The other travels at bearing 235° with a cruising speed of 440 mph. Approximate the distance between the planes after 3 hours of flight.

Question 32

A pilot wishes to fly from Pleasant Hills to Sheldon. She calculates the distances shown using a map, with York for reference since it is due east from Pleasant Hills. What heading should she set for this trip (i.e., what is the measure of angle P)? 91 miles 790 miles

A) 3.8°
B) 4.7°
C) 5.9°
D) 6.6°

Question 33

Solve using the law of cosines (if possible). Round to tenths.
side a = 153 yd
side b = 168 yd
side c = 103 yd

Question 34

Use the following to answer questions :
v =

-Find the acute angle $\theta$ formed by the vector and the nearest x-axis. Round to the nearest tenth of a degree.

Question 35

Use the following to answer questions :
v =
Compute the magnitude of the vector exactly.

Question 36

Vector v₁ is a geometric vector representing a car traveling at 15 mph. Vectors v₂, v₃, and v₄ are vectors representing cars traveling at 10 mph, 20 mph, and 5 mph respectively. Draw these vectors given that v₄ is traveling the same direction and parallel to v₁, while v₂ and v₃ are traveling in the opposite direction and parallel to v₁.

A)
B)
C)
D)

Question 37

Solve the triangle using the law of cosines. Round to tenths.

Question 38

Solve for the unknown part.. Round to one decimal place. b² = (7)² + (6)² - 2(7)(6)cos(67°)

A) b = 7.2
B) b = 8.0
C) b = 8.6
D) b = 9.4

Question 39

Solve for the unknown part. Round to one decimal place. 5² = (9)² + (10)² - 2(9)(10)cos(A)

A) A $\approx$ 29.4°
B) A $\approx$ 29.6°
C) A $\approx$ 29.9°
D) A $\approx$ 30.2°

Question 40

Use the following to answer questions :
Vector v = has initial point (2, 5).
Find the magnitude |v| of the vector.

A) 13
B) 37
C)
D)

Question 41

Find the amount of work required to move an object along the entire length of v with force F. Assume force is in pounds and distance is in feet.
F = ; v =

Question 42

Hal pushes a box full of books 40 ft. If he uses a constant force of 75 pounds, how much work did he do?

Question 43

Find a unit vector pointing in the same direction as the vector v = -2i - 5j.

Question 44

Use the following to answer questions :
u = ; v =
Compute 4u + 5v.

A)
B)
C)
D)

Question 45

Use the following to answer questions :
p = ; q =
Find the angle between the vectors to the nearest tenth of a degree.

A) 165.5°
B) 166.3°
C) 166.7°
D) 168.0°

Question 46

Use the following to answer questions :
p = ; q =
Compute the dot product p • q.

A)
B)
C) -13
D) 11

Question 47

Find the component of u along v (compute comp_vu) for the vectors u and v given. Round to the nearest hundredth. 1475 lbs

Question 48

Find a unit vector pointing in the same direction as the vector v = .

A)
B)
C)
D)

Question 49

Use the following to answer questions :
u = -2i + j; v = 4i + 3j
Compute u + v.

A) i + j
B) 9i - 7j
C) 9i - 6j
D) 8i - 7j

Question 50

Find the component of u along v (compute comp_vu) for the vectors u and v given. Round to the nearest hundredth.

A) 8.60 tons
B) -8.60 tons
C) 12.29 tons
D) -12.29 tons

Question 51

An arrow is shot into the air at an angle of 66° with an initial velocity of 18 ft/sec. Compute the horizontal and vertical components of the representative vector. Round to the nearest tenth.

A) Horizontal component: 7.8 ft/sec; Vertical component: 15.9 ft/sec
B) Horizontal component: 15.9 ft/sec; Vertical component: 7.8 ft/sec
C) Horizontal component: 16.4 ft/sec; Vertical component: 7.3 ft/sec
D) Horizontal component: 7.3 ft/sec; Vertical component: 16.4 ft/sec

Question 52

A baby-sitter pulls some kids in a wagon on a level street. How much work is done if she pulls the wagon 200 feet at a constant force of 45 lbs with the wagon handle making an angle of 34° with the street? Round to the nearest whole number.

A) approximately 4808 ft-lbs
B) approximately 5033 ft-lbs
C) approximately 6529 ft-lbs
D) approximately 7461 ft-lbs

Question 53

The force vectors F₁ = -5i + j and F₂ = -4i - 5j are acting on a common point P. Find an additional force vector so that equilibrium takes place.

Question 54

For the vector below, $\theta$ represents the acute angle formed by the vector and the x-axis. Write the vector in i, j form. Round to the nearest tenth. v in QIV, |v| = 29, $\theta$ = 80°

A) v = 28.6i + 5j
B) v = 28.6i - 5j
C) v = 5i + 28.6j
D) v = 5i - 28.6j

Question 55

Use the following to answer questions :
u = -2i + j; v = 4i + 3j
Compute u - 5v.

A) -16i - 17j
B) -15i - 16j
C) -15i - 17j
D) -16i - 16j

Question 56

Use the following to answer questions :
u = -2i + j; v = 4i + 3j
Compute u - v.

A) -8i + 4j
B) 8i - 4j
C) -10i + 2j
D) 0i + 8j

Question 57

v = (a) Graph the vector.
(b) Write the vector as a linear combination of i and j.
(c) Compute the magnitude of the vector.

Question 58

The force vectors F₁ = and F₂ = are acting on a common point P. Find an additional force vector so that equilibrium takes place.

A)
B)
C)
D)

Question 59

Use the following to answer questions :
u = ; v =
Compute u - 5v.

A)
B)
C)
D)

Question 60

Use the following to answer questions :
u = -2i + j; v = 4i + 3j
Compute 2u + 5v.

A) -17i + 7j
B) -16i + 8j
C) -17i + 8j
D) -16i + 7j

Question 61

Use the following to answer questions :
For a certain AC circuit, R = 35 $\Omega$ , X_L = 20 $\Omega$ and X_C = 8 $\Omega$ , with I = 2 A.

-Find the total voltage across the circuit.

A) 74V
B) 75V
C) 76V
D) 77V

Question 62

Use the following to answer questions :
A projectile is launched from a catapult with initial velocity 350 feet/sec at an angle of 65°.
Find the position of the object after 4 seconds. Round to the nearest hundredth.

A) projectile is about 591.67 ft away and 1268.83 ft high.
B) projectile is about 591.67 ft away and 1012.83 ft high.
C) projectile is about 147.92 ft away and 1268.83 ft high.
D) projectile is about 147.92 ft away and 1012.83 ft high.

Question 63

Use the following to answer questions :
z₁ = 3 - 4i
z₂ = -1 - 3i
z₃ = 4 - i
Express one complex number as the sum of the other two.

A) z₁ = z₂ + z₃
B) z₂ = z₁ + z₃
C) z₃ = z₁ + z₂
D) not possible

Question 64

Use De Moivre's Theorem to compute .

Question 65

Use the following to answer questions :
Compute the product z₁z₂ using the trigonometric form. Write your answer in exact rectangular form.

A)
B)
C)
D)

Question 66

Use the following to answer questions :
z₁ = 3 - 4i
z₂ = -1 - 3i
z₃ = 4 - i
Graph the complex numbers z₁, z₂, and z₃ given.

A)
(Gridlines are spaced one unit apart.)
B)
(Gridlines are spaced one unit apart.)
C)
(Gridlines are spaced one unit apart.)
D)
(Gridlines are spaced one unit apart.)

Question 67

For the complex numbers z₁ = 1 - i and z₂ = -5 + 0i
(a) Find the moduli r₁ and r₂ and the arguments $\theta$ ₁ and $\theta$ ₂.
(b) Compute the quotient in rectangular form.
(c) Find the modulus r and argument $\theta$ of the quotient.
(d) Verify that = r and $\theta$ ₁ - $\theta$ ₂ = $\theta$ .

Question 68

Write the complex number in trigonometric form. Answer in radians using both an exact form and an approximate form, rounding to four decimal places.
-4 + 6i

Question 69

Write the complex number in trigonometric form using degrees. -4 + 4i

A) 4 (cos 45° + i sin 45°)
B) 32(cos 45° + i sin 45°)
C) 4 (cos 135° + i sin 135°)
D) 32(cos 135° + i sin 135°)

Question 70

Use the following to answer questions :
Compute the quotient using the trigonometric form. Write your answer in exact rectangular form.

A) 4
B) -4
C) 4i
D) -4i

Question 71

Use the following to answer questions :
For a certain AC circuit, R = 35 $\Omega$ , X_L = 20 $\Omega$ and X_C = 8 $\Omega$ , with I = 2 A.

-Find the magnitude of Z, the phase angle between current and voltage, and write the result in trigonometric form.

A) 37 cis 68.2°
B) 37 cis 18.9°
C) 38 cis 68.2°
D) 38 cis 18.9°

Question 72

Graph the complex number using its trigonometric form, then convert it to rectangular form.

Question 73

Use the following to answer questions :
A projectile is launched from a catapult with initial velocity 350 feet/sec at an angle of 65°.
Find the time(s) required to reach a height of 200 feet? Round to the nearest hundredth.

A) approximately 0.65 sec.
B) approximately 20.30 sec.
C) approximately 0.65 sec. and 19.17 sec.
D) approximately 0.65 sec. and 20.30 sec.

Question 74

Use the following to answer questions :
u = ; v =
Find the projection of u along v (compute proj_vu). Round to the nearest hundredth as necessary.

A)
B)
C)
D)

Question 75

Use De Moivre's Theorem to compute (4 + 4i)⁴.

A) 1024
B) -1024
C) 1024i
D) -1024i

Question 76

Use De Moivre's Theorem to verify z = -1 + i is a solution to z⁴ - 2z³ - z² + 2z + 10 = 0.

Question 77

Convert the complex number to rectangular form.

A)
B)
C)
D)

Question 78

For the complex numbers z₁ = 3 + 3i and z₂ = -2 - 2i
(a) Find the moduli r₁ and r₂ and the arguments $\theta$ ₁ and $\theta$ ₂.
(b) Compute their product in rectangular form.
(c) Find the modulus r and argument $\theta$ of the product.
(d) Verify that r₁r₂ = r and $\theta$ ₁ + $\theta$ ₂ = $\theta$ .

Question 79

Use the following to answer questions :
Compute the product and the quotient using the trigonometric form. Write your answer in rectangular form. Round to two decimal places as necessary. = 15(cos 72° + i sin 72°), = 2.5(cos 36° + i sin 36°)

Question 80

Use the following to answer questions :
u = ; v =

-Resolve u into vectors u₁ and u₂ where u₁ || v and u₂ $\bot$ v. Round to the nearest hundredth as necessary.

A) u₁ = , u₂ =

B) u₁ = , u₂ =

C) u₁ = , u₂ =

D) u₁ = , u₂ =