Deck 4: Trigonometric Functions of Angles

Question 1

Solve the given triangle. Round angle to one decimal place and side measures to 3 significant digits.

b = 246 yd., c = 377 yd. , a = 426 yd.

Answer 1

$\beta$ = 35.0°, $\gamma$ = 61.4, $\alpha$ = 83.6°

Question 2

Solve the given triangle. Round angle to one decimal place and side measures to 3 significant digits.

b = 63 in., $\gamma$ = 48.4°, a = 56.6 in.

Answer 2

$\beta$ = 72.6°, $\alpha$ = 59.0°, c = 49.4 in.

Question 3

Solve the given triangle. Round angle to one decimal place and side measures to 3 significant digits.

C = 199 yd., a = 232 yd. , b = 219 yd.

A) $\gamma$ = 52.2°, $\alpha$ = 67.2°, $\beta$ = 60.6°
B) $\gamma$ = 60.6°, $\alpha$ = 52.2°, $\beta$ = 67.2°
C) $\gamma$ = 67.2°, $\alpha$ = 60.6°, $\beta$ = 52.2°
D) no solution

Answer 3

$\gamma$ = 52.2°, $\alpha$ = 67.2°, $\beta$ = 60.6°

Question 4

Solve the given triangle. Round angle to one decimal place and side measures to 3 significant digits.

B = 94 ft., $\gamma$ = 45.8°, a = 89.5 ft.

A) $\beta$ = 70.4°, $\alpha$ = 63.8°, c = 71.5 ft.
B) $\beta$ = 63.8°, $\alpha$ = 70.4°, c = 97 ft.
C) $\beta$ = 70.4°, $\alpha$ = 63.8°, c = 106 ft.
D) no solution

Question 5

A plane flew due north at 460 mph for 3.5 hours. A second plane, starting at the same point and at the same time flew southeast at an angle 140° clockwise from due north for 3.5 hours. If, at the end of the 3.5 hours, the two planes were 2550 miles apart, how fast was the second plane traveling?

Question 6

A 26 foot slide leaning against the bottom of a building's window makes a 57° angle with the building. The angle formed with the building with the line of sight from the top of the window to the point on the ground where the slide ends is 39°. How tall is the window? Round to the nearest integer.

Question 7

An airplane door is 5 feet high. If a slide attached to the bottom of the open door is at an angle of 28° with the ground, and the angle formed by the line of sight from where the slide touches the ground to the top of the door is 39°, then how long is the slide?

Question 8

A plane flew due north at 380 mph for 4 hours. A second plane, starting at the same point and at the same time flew southeast at an angle 168° clockwise from due north for 4 hours. If, at the end of the 4 hours, the two planes were 2300 miles apart, how fast was the second plane traveling?

A) 18.3 mph
B) 772 mph
C) 193 mph
D) no solution

Question 9

A 36 foot slide leaning against the bottom of a building's window makes a 52° angle with the building. The angle formed with the building with the line of sight from the top of the window to the point on the ground where the slide ends is 39°. How tall is the window? Round to the nearest integer.

A) 46 feet
B) 13 feet
C) 127 feet
D) no solution

Question 10

An airplane door is 5 feet high. If a slide attached to the bottom of the open door is at an angle of 40° with the ground, and the angle formed by the line of sight from where the slide touches the ground to the top of the door is 50°, then how long is the slide?

A) 19 feet
B) 22 feet
C) 6 feet
D) no solution

Question 11

Find the area of the triangle described. Round to the nearest integer.

b = 370, c = 439, $\gamma$ = 36

Question 12

Find the area of the triangle described. Round to the nearest integer.

c = 428, a = 416 , b = 120

Question 13

A parking lot is to have the shape of a parallelogram that has adjacent sides measuring 158 feet and 277 feet. The angle between the two sides is 43°. What is the area of the parking lot? Round to the nearest integer.

Question 14

A regular heptagon has sides measuring 36 feet. What is its area? (Hint: The measure of an angle of a regular n-gon is equal to 180(n - 2) divided by the number of sides, n.) Round to one decimal place.

Question 15

Find the area of the triangle described. Round to the nearest integer.

A = 274, b = 257, $\beta$ = 29°

A) 17,070
B) 34,139
C) 23,366
D) no solution

Question 16

Find the area of the triangle described. Round to the nearest integer.

B = 493, c = 402 , a = 338

A) 67,441
B) 4,548,335,944
C) 823,777
D) 815,453

Question 17

A parking lot is to have the shape of a parallelogram that has adjacent sides measuring 281 feet and 253 feet. The angle between the two sides is 57°. What is the area of the parking lot? Round to the nearest integer.

A) 59,624 square feet
B) 29,812 square feet
C) 31,008 square feet
D) 38,720 square feet

Question 18

A regular decagon has sides measuring 38 inches. What is its area? (Hint: The measure of an angle of a regular n-gon is equal to 180(n - 2) divided by the number of sides, n.) Round to one decimal place.

A) 11110.4 square inches
B) 3795.8 square inches
C) 4968.7 square inches
D) 11682.2 square inches

Question 19

Peg and Meg live five miles apart. The school that they attend lies on a street that makes a 34° angle with the street connecting their houses when measured from Peg's house. The street connecting Meg's house and the school makes a 38° angle with the street connecting them. How far is it from Peg's house to the school?

Question 20

Decide whether the Law of Cosines is needed to solve this triangle.

Given: sides a and b, and angle $\gamma$

A) Do NOT need Law of Cosines
B) Do need Law of Cosines

Question 21

Find the area of the triangle described. Round to three decimal places.

b =

, c =

, $\gamma$ = 54

Question 22

Find the area of the triangle described. Round to one decimal place.

a =

, b = 38 , c = 39

Question 23

Some very destructive beetles have made their way into a forest preserve. The rangers are trying to keep track of their spread and how well preventative measures are working. In a triangular area that is 49.9 miles on one side, 43.5 miles on another and 57.0 miles on the third, what is the total area the rangers are covering? Round to one decimal place.

Question 24

A real estate agent needs to determine the area of a triangular lot. Two sides of the lot are 130 feet and 100 feet. The angle between the two measured sides is 79°. What is the area of the lot? Round to one decimal place.

Question 25

Big Augie's Pizza now offers equilateral triangle shaped pizzas. How long must the sides of the pizza be to the nearest inch to equal the size of a round pizza with a 17 inch diameter?

Question 26

Classify the triangle as AAS, SAS, SSA, SAS, or SSS given the following information.
b, c $\alpha$

Question 27

Classify the triangle as AAS, SAS, SSA, SAS, or SSS given the following information.
c $\gamma$

Question 28

Solve the given triangle. Round angle to one decimal place and side measures to 3 significant digits.

b = 142 yd., $\beta$ = 54.2°, $\gamma$ = 60.4°

Question 29

Solve the given triangle. Round angle to one decimal place and side measures to 3 significant digits.

b = 217 yd., $\gamma$ = 54.2°, $\alpha$ = 37.0°

Question 30

Solve the given triangle. Round angles to one decimal place and side measure to 3 significant figures.

c = 151 cm., a = 55 cm., $\gamma$ = 69.0°

Question 31

Solve the given triangle. Round angles to one decimal place and side measures to 3 significant figures.

b = 67 yd., c = 70.3 yd., $\beta$ = 70.0°

Question 32

Solve the given triangle and round to one decimal place.

A = 151.0 yd., $\alpha$ = 69.0°, $\beta$ = 63.4°

A) $\gamma$ = 47.6°^, b = 144.6 yd. , c = 119.4 yd.
B) $\gamma$ = 47.6°^, b = 119.4 yd. , c = 144.6 yd.
C) $\gamma$ = 47.6°^, b = 157.7 yd. , c = 124.7 yd.
D) $\gamma$ = 47.6°^, b = 190.9 yd. , c = 182.8 yd.

Question 33

Solve the given triangle and round to one decimal place.

B = 57.0 cm., $\gamma$ = 61.8°, $\alpha$ = 63.4°

A) $\gamma$ = 54.8°^, c = 61.5 cm. , a = 62.4 cm.
B) $\gamma$ = 54.8°^, c = 62.4 cm. , a = 61.5 cm.
C) $\gamma$ = 54.8°^, c = 52.9 cm. , a = 57.8 cm.
D) $\gamma$ = 54.8°^, c = 52.1 cm. , a = 57.0 cm.

Question 34

Solve the given triangle and round to one decimal place.

B = 199.0 cm., c = 45.0 cm., $\beta$ = 40.0°

A) $\gamma$ = 8.4°, $\alpha$ = 131.6°, a = 231.4 cm.
B) $\gamma$ = 131.6°, $\alpha$ = 8.4°, a = 231.4 cm.
C) $\gamma$ = 8.4°, $\alpha$ = 131.6°, a = 171.2 cm.
D) no solution

Question 35

Solve the given triangle and round to one decimal place.

B = 33.0 in., c = 35.8 in., $\beta$ = 30.0°

A) Case one: $\gamma$ ₁ = 32.8°, $\alpha$ ₁ = 117.2°, a₁ = 58.7 in.
Case two: $\gamma$ ₂ = 147.2°, $\alpha$ ₂ = 2.8°, a₂ = 3.3 in.
B) Case one: $\gamma$ ₁ = 32.8°, $\alpha$ ₁ = 117.2°, a₁ = 58.7 in.
Case two: $\gamma$ ₂ = 2.8°, $\alpha$ ₂ = 147.2°, a₂ = 3.3 in.
C) $\gamma$ = 32.8°, $\alpha$ = 117.2°, a = 58.7 in. and no ambiguous case
D) no solution

Question 36

A hot air balloon is sighted at the same time by two friends who are 2.2 miles apart on the same side of the balloon. The angles of elevation from the two friends are 11.5° and 18°. How high is the balloon? Round to one decimal place.

Question 37

A tracking station has two telescopes that are 1.6 miles apart. The telescopes can lock onto a rocket after it is launched and record the angles of elevation to the rocket. If the angles of elevation from telescope A and B are 27° and 56°, respectively, then how far is the rocket from telescope A? Round to one decimal place.

Question 38

An engineer wants to construct a bridge across a fast moving river. Using a straight line segment between two points that are 187 feet apart along his side of the river, he measures the angles formed when sighting the point on the other side where he wants to have the bridge end. If the angles formed at points A and B are 68° and 34°, respectively, how far is it from point B to the point on the other side of the river? Round to one decimal place.

Question 39

A hot air balloon is sighted at the same time by two friends who are 2.5 miles apart on the same side of the balloon. The angles of elevation from the two friends are 20.5° and 28°. How high is the balloon? Round to one decimal place.

A) 3.1 miles
B) 0.2 mile
C) 0.4 mile
D) no solution

Question 40

A tracking station has two telescopes that are 1.7 miles apart. The telescopes can lock onto a rocket after it is launched and record the angles of elevation to the rocket. If the angles of elevation from telescope A and B are 22.5° and 44°, respectively, then how far is the rocket from telescope B? Round to one decimal place.

A) 1.8 miles
B) 3.2 miles
C) 1.6 miles
D) no solution

Question 41

An engineer wants to construct a bridge across a fast moving river. Using a straight line segment between two points that are 95 feet apart along his side of the river, he measures the angles formed when sighting the point on the other side where he wants to have the bridge end. If the angles formed at points A and B are 78° and 37°, respectively, how far is it from point A to the point on the other side of the river? Round to one decimal place.

A) 63.1 feet
B) 102.5 feet
C) 143.1 feet
D) no solution

Question 42

The terminal side of an angle θ in standard position passes through the point

. Calculate the exact value of sin θ.

Question 43

The terminal side of an angle θ in standard position passes through the point

. Calculate the exact value of csc θ.

Question 44

The terminal side of an angle θ in standard position passes through the point

. Calculate the exact value of cot θ.

Question 45

The terminal side of an angle θ in standard position passes through the point

. Calculate the exact value of sin θ.

Question 46

The terminal side of an angle θ in standard position passes through the point

. Calculate the exact value of cot θ.

Question 47

The terminal side of an angle θ in standard position passes through the point

. Calculate the exact value of sec θ.

Question 48

The terminal side of an angle θ in standard position passes through the point

. Calculate the value of sin θ.

Question 49

The terminal side of an angle θ in standard position passes through the point

. Calculate the value of cos θ.

Question 50

The terminal side of an angle θ in standard position passes through the point

. Calculate the value of tan θ.

Question 51

The terminal side of an angle θ in standard position passes through the point

. Calculate the value of cot θ.

Question 52

The terminal side of an angle θ in standard position passes through the point

. Calculate the value of csc θ.

Question 53

The terminal side of an angle θ in standard position passes through the point

. Calculate the value of sec θ.

Question 54

The terminal side of an angle θ in standard position passes through the point (-2, -1). Calculate the value of sinθ. Round your answer to four decimal places.

Question 55

The terminal side of an angle θ in standard position passes through the point (-6, 1). Calculate the value of cosθ. Round your answer to four decimal places.

Question 56

The terminal side of an angle θ in standard position passes through the point (2, 5). Calculate the value of tanθ. Round your answer to four decimal places.

Question 57

The terminal side of an angle θ in standard position passes through the point (-4, 6). Calculate the value of cotθ. Round your answer to four decimal places.

Question 58

The terminal side of an angle θ in standard position passes through the point (7, -6). Calculate the value of cscθ. Round your answer to four decimal places.

Question 59

The terminal side of an angle θ in standard position passes through the point (2, 1). Calculate the value of secθ. Round your answer to four decimal places.

Question 60

Calculate cot $\theta$ if $\theta$ is in standard position and has a measure of 225°.

A) 0
B) -1
C) 1
D)

Question 61

Calculate cscθ if θ is in standard position and has a measure of 1890°.

Question 62

A right triangle is drawn in QI with one leg on the x-axis and its hypotenuse on the terminal side of

$\theta$ drawn in standard position. If tan $\theta$ =

, then what is sin $\theta$ ?

A)
B)
C)
D)

Question 63

A right triangle is drawn in QI with one leg on the x-axis and its hypotenuse on the terminal side of

$\theta$ drawn in standard position. If tan $\theta$ =

, then what is cos $\theta$ ?

A)
B)
C)
D)

Question 64

A right triangle is drawn in QI with one leg on the x-axis and its hypotenuse on the terminal side of

$\theta$ drawn in standard position. If sin $\theta$ =

, then what is tan $\theta$ ?

A)
B)
C)
D)

Question 65

A right triangle is drawn in QI with one leg on the x-axis and its hypotenuse on the terminal side of

θ drawn in standard position. If tanθ =

, then what is sinθ?

Question 66

A right triangle is drawn in QI with one leg on the x-axis and its hypotenuse on the terminal side of

θ drawn in standard position. If tanθ =

, then what is cosθ?

Question 67

A right triangle is drawn in QI with one leg on the x-axis and its hypotenuse on the terminal side of

θ drawn in standard position. If sin θ =

, then what is tanθ?

Question 68

Indicate the quadrant in which the terminal side of $\theta$ must lie in order for the following to be true.
Tan $\theta$ is positive and cos $\theta$ is negative.

A) Q1
B) QII
C) QIII
D) QIV

Question 69

Indicate the quadrant in which the terminal side of $\theta$ must lie in order for the following to be true:
Cos $\theta$ is positive and tan $\theta$ is negative.

A) Q1
B) QII
C) QIII
D) QIV

Question 70

Indicate the quadrant in which the terminal side of θ must lie in order for the following to be true:
sinθ is positive and tanθ is negative.

Question 71

Indicate the quadrant in which the terminal side of θ must lie in order for the following to be true:
tanθ is positive and cosθ is negative.

Question 72

If tan $\theta$ =

, and the terminal side of $\theta$ lies in quadrant II, find sin $\theta$ .

A)
B)
C)
D)

Question 73

If tan $\theta$ =

, and the terminal side of $\theta$ lies in quadrant II, find cos $\theta$ .

A)
B)
C)
D)

Question 74

If cos $\theta$ =

, and the terminal side of $\theta$ lies in quadrant I, find sin $\theta$ .

A)
B)
C)
D)

Question 75

If cot $\theta$ =

, and the terminal side of $\theta$ lies in quadrant II, find sec $\theta$ .

A)
B)
C)
D)

Question 76

If sec $\theta$ =

, and the terminal side of $\theta$ lies in quadrant I, find sin $\theta$ .

A)
B)
C)
D)

Question 77

If tan $\theta$ =

, and the terminal side of $\theta$ lies in quadrant III, find csc $\theta$ .

A)
B)
C)
D)

Question 78

Indicate the quadrant in which the terminal side of θ must lie in order for the following to be true.
tan θ is positive and sin θ is negative.

Question 79

If sinθ =

, and the terminal side of θ lies in quadrant IV, find cosθ.

Question 80

If cos θ =

, and the terminal side of θ lies in quadrant I, find sin θ.