# Precalculus with Limits

## Educators

Problem 1

Vocabulary: Fill in the blanks.
An_____triangle is a triangle that has no right angle.

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Problem 2

Vocabulary: Fill in the blanks.
For triangle $A B C,$ the Law of Sines is $\frac{a}{\sin A}=$______ $=\frac{c}{\sin C}$

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Problem 3

Vocabulary: Fill in the blanks.
Two ________ and one ________ determine a unique triangle.

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Problem 4

Vocabulary: Fill in the blanks.
The area of an oblique triangle is $\frac{1}{2} b c \sin A=\frac{1}{2} a b \sin C=$ ________.

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Problem 5

Using the Law of Sines. Use the Law of Sines to solve the triangle. Round your answers to two decimal places.

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Problem 6

Using the Law of Sines. Use the Law of Sines to solve the triangle. Round your answers to two decimal places.

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Problem 7

Using the Law of Sines. Use the Law of Sines to solve the triangle. Round your answers to two decimal places.

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Problem 8

Using the Law of Sines. Use the Law of Sines to solve the triangle. Round your answers to two decimal places.

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Problem 9

Using the Law of Sines. Use the Law of Sines to solve the triangle. Round your answers to two decimal places.

$$A=102.4^{\circ}, \quad C=16.7^{\circ}, \quad a=21.6$$

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Problem 10

Using the Law of Sines. Use the Law of Sines to solve the triangle. Round your answers to two decimal places.

$$A=24.3^{\circ}, \quad C=54.6^{\circ}, \quad c=2.68$$

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Problem 11

Using the Law of Sines. Use the Law of Sines to solve the triangle. Round your answers to two decimal places.

$$A=83^{\circ} 20^{\prime}, \quad C=54.6^{\circ}, \quad c=18.1$$

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Problem 12

Using the Law of Sines. Use the Law of Sines to solve the triangle. Round your answers to two decimal places.

$$A=5^{\circ} 40^{\prime}, \quad B=8^{\circ} 15^{\prime}, \quad b=4.8$$

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Problem 13

Using the Law of Sines. Use the Law of Sines to solve the triangle. Round your answers to two decimal places.

$$A=35^{\circ}, \quad B=65^{\circ}, \quad c=10$$

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Problem 14

Using the Law of Sines. Use the Law of Sines to solve the triangle. Round your answers to two decimal places.

$$A=120^{\circ}, \quad B=45^{\circ}, \quad c=16$$

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Problem 15

Using the Law of Sines. Use the Law of Sines to solve the triangle. Round your answers to two decimal places.

$$A=55^{\circ}, \quad B=42^{\circ}, \quad c=\frac{3}{4}$$

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Problem 16

Using the Law of Sines. Use the Law of Sines to solve the triangle. Round your answers to two decimal places.

$$B=28^{\circ}, \quad C=104^{\circ}, \quad a=3 \frac{5}{8}$$

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Problem 17

Using the Law of Sines. Use the Law of Sines to solve the triangle. Round your answers to two decimal places.

$$A=36^{\circ}, \quad a=8, \quad b=5$$

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Problem 18

Using the Law of Sines. Use the Law of Sines to solve the triangle. Round your answers to two decimal places.

$$A=60^{\circ}, \quad a=9, \quad c=10$$

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Problem 19

Using the Law of Sines. Use the Law of Sines to solve the triangle. Round your answers to two decimal places.

$$B=15^{\circ} 30^{\prime}, \quad a=4.5, \quad b=6.8$$

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Problem 20

Using the Law of Sines. Use the Law of Sines to solve the triangle. Round your answers to two decimal places.

$$B=2^{\circ} 45^{\prime}, \quad b=6.2, \quad c=5.8$$

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Problem 21

Using the Law of Sines. Use the Law of Sines to solve the triangle. Round your answers to two decimal places.

$$A=145^{\circ}, \quad a=14, \quad b=4$$

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Problem 22

Using the Law of Sines. Use the Law of Sines to solve the triangle. Round your answers to two decimal places.

$$A=100^{\circ}, \quad a=125, \quad c=10$$

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Problem 23

Using the Law of Sines. Use the Law of Sines to solve the triangle. Round your answers to two decimal places.

$$A=110^{\circ} 15^{\prime}, \quad a=48, \quad b=16$$

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Problem 24

Using the Law of Sines. Use the Law of Sines to solve the triangle. Round your answers to two decimal places.

$$C=95.20^{\circ}, \quad a=35, \quad c=50$$

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Problem 25

Using the Law of Sines. Use the Law of Sines to solve (if possible) the triangle. If two solutions exist, find both. Round your answers to two decimal places.

$$A=110^{\circ}, \quad a=125, \quad b=100$$

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Problem 26

Using the Law of Sines. Use the Law of Sines to solve (if possible) the triangle. If two solutions exist, find both. Round your answers to two decimal places.

$$A=110^{\circ}, \quad a=125, \quad b=200$$

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Problem 27

Using the Law of Sines. Use the Law of Sines to solve (if possible) the triangle. If two solutions exist, find both. Round your answers to two decimal places.

$$A=76^{\circ}, \quad a=18, \quad b=20$$

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Problem 28

Using the Law of Sines. Use the Law of Sines to solve (if possible) the triangle. If two solutions exist, find both. Round your answers to two decimal places.

$$A=76^{\circ}, \quad a=34, \quad b=21$$

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Problem 29

Using the Law of Sines. Use the Law of Sines to solve (if possible) the triangle. If two solutions exist, find both. Round your answers to two decimal places.

$$A=58^{\circ}, \quad a=11.4, \quad b=12.8$$

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Problem 30

Using the Law of Sines. Use the Law of Sines to solve (if possible) the triangle. If two solutions exist, find both. Round your answers to two decimal places.

$$A=58^{\circ}, \quad a=4.5, \quad b=12.8$$

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Problem 31

In Exercises 25-34, use the Law of Sines to solve (if possible) the triangle. If two solutions exist, find both. Round your answers to two decimal places.
$$A=120^{\circ}, \quad a=b=25$$c

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Problem 32

In Exercises 25-34, use the Law of Sines to solve (if possible) the triangle. If two solutions exist, find both. Round your answers to two decimal places.
$$A=120^{\circ}, \quad a=25, \quad b=24$$

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Problem 33

In Exercises 25-34, use the Law of Sines to solve (if possible) the triangle. If two solutions exist, find both. Round your answers to two decimal places.
$$A=45^{\circ}, \quad a=b=1$$

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Problem 34

In Exercises 25-34, use the Law of Sines to solve (if possible) the triangle. If two solutions exist, find both. Round your answers to two decimal places.
$$A=25^{\circ} 4^{\prime} ; \quad a=9.5, \quad b=22$$

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Problem 35

Finding the Area of a Triangle In Exercises $39-46$ find the area of the triangle having the indicated $$A=5^{1} 15 ; \quad b=4.5, \quad c=22$$

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Problem 35

Using the Law of Sines In Exercises $35-38$ , find values for $b$ such that the triangle has (a) one solution, (b) two solutions, and (c) no solution.
$$A=36^{\circ}, \quad a=5$$

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Problem 36

Using the Law of Sines In Exercises $35-38$ , find values for $b$ such that the triangle has (a) one solution, (b) two solutions, and (c) no solution.
$$A=60^{\circ}, \quad a=10$$

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Problem 37

Using the Law of Sines In Exercises $35-38$ , find values for $b$ such that the triangle has (a) one solution, (b) two solutions, and (c) no solution.
$$A=10^{\circ}, \quad a=10.8$$

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Problem 38

Using the Law of Sines In Exercises $35-38$ , find values for $b$ such that the triangle has (a) one solution, (b) two solutions, and (c) no solution.
$$A=88^{\circ}, \quad a=315.6$$

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Problem 39

Finding the Area of a Triangle In Exercises $39-46$ find the area of the triangle having the indicated angle and sides. $$C=120^{\circ}, \quad a=4, \quad b=6$$

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Problem 40

Finding the Area of a Triangle In Exercises $39-46$ find the area of the triangle having the indicated $$B=130^{\circ}, \quad a=62, \quad c=20$$

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Problem 41

Finding the Area of a Triangle In Exercises $39-46$ find the area of the triangle having the indicated $$A=150^{\circ}, \quad b=8, \quad c=10$$

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Problem 42

Finding the Area of a Triangle In Exercises $39-46$ find the area of the triangle having the indicated $$C=170^{\circ}, \quad a=14, \quad b=24$$

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Problem 43

Finding the Area of a Triangle In Exercises $39-46$ find the area of the triangle having the indicated $$A=43^{\circ} 45^{\prime}, \quad b=57, \quad c=85$$

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Problem 44

Finding the Area of a Triangle In Exercises $39-46$ find the area of the triangle having the indicated $$A=5^{1} 15 ; \quad b=4.5, \quad c=22$$

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Problem 45

Finding the Area of a Triangle In Exercises $39-46$ find the area of the triangle having the indicated $$A=5^{1} 15 ; \quad b=4.5, \quad c=22$$

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Problem 45

Finding the Area of a Triangle In Exercises $39-46$ find the area of the triangle having the indicated $$B=72^{\circ} 30^{\prime}, \quad a=105, \quad c=64$$

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Problem 46

Finding the Area of a Triangle In Exercises $39-46$ find the area of the triangle having the indicated $$C=84^{\circ} 30^{\prime}, \quad a=16, \quad b=20$$

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Problem 47

Height Because of prevailing winds, a tree grew so that it was leaning $4^{\circ}$ from the vertical. At a point 40 meters from the tree, the angle of elevation to the top of the tree is $30^{\circ}$ (see figure). Find the height $h$ of the tree.

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Problem 48

Height A flagpole at a right angle to the horizontal is located on a slope that makes an angle of $12^{\circ}$ with the horizontal. The flagpole's shadow is 16 meters long and points directly up the slope. The angle of elevation from the tip of the shadow to the sun is $20^{\circ} .$
(a) Draw a triangle to represent the situation. Show the known quantities on the triangle and use a variable to indicate the height of the flagpole. (b) Write an equation that can be used to find the height
of the flagpole. (c) Find the height of the flagpole.

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Problem 49

Angle of Elevation A 10 -meter utility pole casts a 17 -meter shadow directly down a slope when the angle of elevation of the sun is $42^{\circ}$ (see figure). Find $\theta,$ the angle of elevation of the ground.

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Problem 50

Bridge Design A bridge is to be built across a small lake from a gazebo to a dock (see figure). The bearing from the gazebo to the dock is $\ 41^{\circ} \mathrm{W}$ . From a tree 100 meters from the gazebo, the bearings to the gazebo and the dock are $\ 74^{\circ}$ E and $S 28^{\circ}$ E, respectively. Find the distance from the gazebo to the dock.

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Problem 51

Flight Path A plane flies 500 kilometers with a bearing of $316^{\circ}$ from Naples to Elgin (see figure). The plane then flies 720 kilometers from Elgin to Canton (Canton is due west of Naples). Find the bearing of the flight from Elgin to Canton

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Problem 52

Locating a Fire The bearing from the Pine Knob fire tower to the Colt Station fire tower is $N 65^{\circ} \mathrm{E},$ and the two towers are 30 kilometers apart. A fire spotted by rangers in each tower has a bearing of $\mathrm{N} 80^{\circ} \mathrm{E}$ from Pine Knob and $\mathrm{S} 70^{\circ} \mathrm{E}$ from Colt Station ( see figure). Find the distance of the fire from each tower.

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Problem 53

A boat is sailing due east parallel to the shoreline at a speed of 10 miles per hour. At a given time, the bearing to the lighthouse is $S 70^{\circ} \mathrm{E}$and 15 minutes later
the bearing is $S 63^{\circ}$ E (see figure). The lighthouse is located at the shoreline. What is the distance from the boat to the shoreline?

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Problem 54

Altitude The angles of elevation to an airplane from
two points $A$ and $B$ on level ground are $55^{\circ}$ and $72^{\circ}$ ,
respectively. The points $A$ and $B$ are 2.2 miles apart, and
the airplane is east of both points in the same vertical
plane. Find the altitude of the plane.

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Problem 55

Distance The angles of elevation $\theta$ and $\phi$ to an
airplane from the airport control tower and from an
observation post 2 miles away are being continuously
monitored (see figure). Write an equation giving the
distance $d$ between the plane and observation post in
terms of $\theta$ and $\phi .$

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Problem 56

The Leaning Tower of Pisa The Leaning Tower
of Pisa in Italy leans because it was built on unstable
soil-a mixture of clay, sand, and water. The tower is
approximately 58.36 meters tall from its foundation
(see figure). The top of the tower leans about 5.45 meters
off center.

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Problem 56

The Leaning Tower of Pisa The Leaning Tower
of Pisa in Italy leans because it was built on unstable
soil-a mixture of clay, sand, and water. The tower is
approximately 58.36 meters tall from its foundation
(see figure). The top of the tower leans about 5.45 meters
off center.
(a) Find the angle of lean $\alpha$ of the tower.
(b) Write $\beta$ as a function of $d$ and $\theta,$ where $\theta$ is the
angle of elevation to the sun.
(c) Use the Law of Sines to write an equation for the
length $d$ of the shadow cast by the tower in terms of
$\theta$ ) Use a graphing utility to complete the table.

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Problem 57

True or False? In Exercises 57-59, determine whether
If a triangle contains an obtuse angle, then it must be
oblique.

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Problem 58

True or False? In Exercises 57-59, determine whether
Two angles and one side of a triangle do not necessarily
determine a unique triangle.

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Problem 59

True or False? In Exercises 57-59, determine whether
If three sides or three angles of an oblique triangle are
known, then the triangle can be solved.

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Problem 60

Graphical and Numerical Analysis In the figure,
$\alpha$ and $\beta$ are positive angles.
(a) Write $\alpha$ as a function of $\beta$
(b) Use a graphing utility to graph the function in
part (a). Determine its domain and range.
(c) Use the result of part (a) to write $c$ as a function
of $\beta .$
(d) Use the graphing utility to graph the function in
part (c). Determine its domain and range.
(e) Complete the table. What can you infer?

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Problem 61

Graphical Analysis
(a) Write the area $A$ of the shaded region in the figure
as a function of $\theta$ .
(b) Use a graphing utility to graph the function.
(c) Determine the domain of the eight-centimeter line
degment would affect the area of the region and the
domain of the function.

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Problem 62

HOW DO YOU SEE IT? In the figure, a
triangle is to be formed by drawing a line
segment of length $a$ from $(4,3)$ to the positive
$x$ -axis. For what value(s) of a can you form
(a) one triangle, (b) two triangles, and (c) no