Deck 6: Trigonometric Functions: Unit Circle Approach

Question 1

Two cities in the western hemisphere lie approximately on the same meridian. Guatemala city has latitude of $15.5 ^ { \circ }$ N and Baton Rouge, Louisiana $30.5 ^ { \circ }$ N. Find the distance between these two cities. (The radius of the earth is $3960$ miles.)

Answer 1

$\theta = 30.5 ^ { \circ } - 15.5 ^ { \circ } = 15 ^ { \circ } = 15 ^ { \circ } \left( \frac { \pi } { 180 ^ { \circ } } \right)$ radians $= \frac { \pi } { 12 }$ .
$s = r \theta = ( 3960 ) \left( \frac { \pi} { 12 } \right) = 330 \pi \approx 1036.7$ miles.

Question 2

Find the area of a sector with central angle $3 \pi / 2$ in a circle with radius $5$ m.

Answer 2

$A = \frac { 1 } { 2 } \cdot 5 ^ { 2 } \cdot \frac { 3 \pi } { 2 } = \frac { 75 \pi } { 4 } \mathrm {~m} ^ { 2 }$

Question 3

Find the radian measure that corresponds to each degree measure.
a) $330 ^ { \circ }$
b) $- 225 ^ { \circ }$

Answer 3

(a) $\frac { 11 \pi } { 6 }$

b) $- \frac { 5 \pi } { 4 }$

Question 4

Rewrite the expression as an algebraic expression in x.

Question 5

Solve for $x$ correct to one decimal place.

Question 6

Use the Law of Sines to solve for all possible triangles that satisfy the given conditions.

Question 7

Find the exact value of the expression.

Question 8

Find the degree measures that correspond to the radian measures
and
. Write your answer(s) to one decimal place.

Question 9

Find $\sec \theta$ and $\csc \theta$

Question 10

Find the values of $\sec \theta$ and $\cot \theta$ if the point $( - 3,4 )$ is on the terminal side of $\theta$ .

Question 11

A 86-ft tree casts a shadow that is 110 ft long. What is the angle of elevation of the sun? Round the answer to two decimal places.

Question 12

Find the acute angle that is formed by the line
and the x-axis.

Question 13

Find the reference angle for the given angles
a) $- 202 ^ { \circ }$

b) $\frac { 19 \pi } { 6 }$

Question 14

Find $\sec \theta$ , and $\csc \theta$ given that $\cot \theta = - 4 / 3$ and $\operatorname { sect } > 0$ .

Question 15

Find the area of a triangle with sides of length 12 and 22 and included angle $12 ^ { \circ }$ .

Question 16

Find all angles $\theta$ between $0$ and $180 ^ { \circ }$ satisfying the given equation. Round all answers to two decimal places. $\sin \theta = 0.7$

Question 17

Find an angle between $0$ and $360 ^ { \circ }$ that is coterminal with each angle given
a) $362 ^ { \circ }$
b) $- 870 ^ { \circ }$
c) $1170 ^ { \circ }$

Question 18

Find the measure of the central angle $\theta$ in a circle of radius 6 ft if the angle is subtended by an arc of length 7 ft.

Question 19

How many revolutions will a bicycle wheel of diameter $25$ inches make as the bicycle travels a distance of $3$ miles?

Question 20

A surveyor stands at a point $1.5$ miles away from the base of a radio tower. He measures the angle of elevation to the top of the tower to be $7.5 ^ { \circ }$ . Find the height of the tower to the nearest foot.

Question 21

Find the exact value of the expression.

Question 22

Find
and
.

Question 23

Find the indicated angle $\theta$ .

Question 24

Find the angle $\theta$ in the triangle. Round the answer to one decimal place.

Question 25

Find the side labeled x.

Question 26

Find the area of a triangle with sides of length 12 and 22 and included angle $10 ^ { \circ }$ . Round the answer to one decimal place.

Question 27

How many revolutions will a bicycle wheel of diameter $26$ inches make as the bicycle travels a distance of $4$ miles?

Question 28

Solve the triangle.

Question 29

Solve for $x$ correct to one decimal place.

Question 30

To find the distance across a river a surveyor chooses points $A$ and $B$ , $100$ m apart and a reference point $C$ on the other side of the river. Approximate the distance, $b$ between $A$ and $C$ given that $\angle \mathrm { A } = 93 ^ { \circ }$ and $\angle B = 62 ^ { \circ }$ .

Question 31

Land on a small Caribbean island is valued at $\$ 45$ per square foot. What is the value of a triangular lot with sides of lengths $80$ ft, $50$ ft, and $120$ ft?

Question 32

Find the measure of the central angle $\theta$ in a circle of radius 8 ft if the angle is subtended by an arc of length 7 ft.

Question 33

Two cities in the western hemisphere lie approximately on the same meridian. Guatemala city has latitude of $15.5 ^ { \circ }$ N and Baton Rouge, Louisiana $30.5 ^ { \circ }$ N. Find the distance between these two cities. (The radius of the earth is $3960$
miles.)

Question 34

Find the length of an arc of a circle of radius $12$
ft if the arc subtends a central angle of $135 ^ { \circ }$ .

Question 35

Find an angle between $0$ and $2 \pi$ that is coterminal with each angle given.
a. $\frac { 17 \pi } { 2 }$
b. $- \frac { 15 \pi } { 4 }$
c. $85 \pi$

Question 36

From the top of a $250$ ft lighthouse, the angle of depression to a ship in the ocean is $18 ^ { \circ }$ . How far is the ship from the base of the lighthouse?

Question 37

A pilot sets out from an airport and heads in the direction N $30 ^ { \circ }$ W, flying at a constant speed of $305$ mi/h. Forty-five minutes later the pilot makes a course and speed correction and now heads in the direction N $50 ^ { \circ }$ W and reduces her speed to $175$ mi/h. Half an hour later, engine trouble forces her to make an emergency landing. Find the distance between the airport and the final landing point.

Question 38

Land in downtown Stevensville is valued at $\$ 20$ a square foot. What is the value of a triangular lot with sides of lengths 112ft, 148ft, and 190 ft?

Question 39

Find the area of a sector with central angle $5 \pi / 4$ in a circle with radius $0.5$ m.

Question 40

Find the radian measures that correspond to the degree measures $- 300 ^ { \circ }$
and $135 ^ { \circ }$ .

Question 41

Find the radian measure that corresponds to each degree measure. ,

A)
B)
C)
D)
E)

Question 42

Find the area of a sector with central angle $120 ^ { \circ }$ in a circle with radius $12$ M

A) $12 \pi \mathrm { m } ^ { 2 }$
B) $24 \mathrm {~m} ^ { 2 }$
C) $36 \mathrm {~m} \mathrm {} \mathrm { m } ^ { 2 }$
D) $48 \mathrm {~m} \mathrm {~m} ^ { 2 }$
E) $2 \pi / 3 m ^ { 2 }$

Question 43

Two cities in the western hemisphere lie approximately on the same meridian. City A has latitude of N and City B,
N) Find the distance between these two cities. (The radius of the earth is
Miles.)

A)
B)
C)
D)
E)

Question 44

Find an angle between $0$ and $2 \pi$ That is coterminal with each angle given.
a. $\frac { 17 \pi } { 2 }$
b. $- \frac { 15 \pi } { 4 }$
c. $85 \pi$

A) $\pi / 2 , \pi / 4 , \pi$
B) $m / 2 , - \pi / 4,0$
C) $3 \pi / 2 , \pi / 4 , \pi$
D) $\pi , 3 \pi / 4 , \pi / 2$
E) $- \pi / 2 , - \pi / 4 , - \pi$

Question 45

Find the side labeled x.

Question 46

Find and
)

A)
B)
C)
D)
E)

Question 47

A boy is riding a scooter whose wheels are inches in diameter. If the wheels rotate at
Rpm, find the speed at which he is traveling in mi/h

A)
B)
C)
D)
E)

Question 48

Find the indicated angle $\theta$ .

Question 49

Find the length of an arc of a circle of radius ft if the arc subtends a central angle of

A)
B)
C)
D)
E)

Question 50

How many revolutions will a bicycle wheel of diameter $26$ inches make as the bicycle travels a distance of $4$ Miles?

A) $2,193$
B) $1,760$
C) $3,103$
D) $3,806$
E) $5,280$

Question 51

Use the Law of Sines to solve for all possible triangles that satisfy the given conditions.

Question 52

To find the distance across a river a surveyor chooses points $A$ and $B$ , $50$ m apart and a reference point $C$ on the opposite side of the river. Approximate the distance, $b$ , between $A$ and $C$ given that $\angle \mathrm { A } = 54 ^ { \circ }$ and $\angle B = 77 ^ { \circ }$ .

Question 53

Find $\sin \theta$ and $\cos \theta$ given that $\sec \theta = - 2$ and $\tan \theta > 0$ .

Question 54

Solve for $x$ correct to one decimal place.

A) $173.2$
B) $230.9$
C) $57.7$
D) $57.9$
E) $114.1$

Question 55

Use Heron's formula to find the area of the triangle with sides of length 10, 12, and 20.

Question 56

Find the values of $\sin \theta$ and $\tan \theta$ if the point $( - 3,4 )$ is on the terminal side of $\theta$ .

Question 57

Find the obtuse angle that is formed by the line
and the x-axis.

Question 58

Find the exact value of the expression.

Question 59

A plane leaves the airport at 4:00 p.m. and heads in the direction N $10 ^ { \circ }$ E at a constant speed of $250$ mi/h. Forty-five minutes later, a jet leaves the same airport and flies N $60 ^ { \circ }$ W at a constant speed of $450$ mi/h. Find the distance between the plane and the jet at 5:30p.m.

Question 60

Find the measure of the central angle $\theta$ in a circle of radius 13 ft if the angle is subtended by an arc of length 7 ft

A) $\frac { 7 } { 13 }$

B) $\frac { 7 \pi } { 13 }$

C) $\frac { 7 } { 13 \pi }$

D) $\frac { \pi } { 13 }$

E) $\frac { 13 \pi } { 7 }$

Question 61

Find the exact value.

A)
B)
C)
D)
E)

Question 62

Find the indicated angle $\theta$ .

A) $16 ^ { \circ }$
B) $56 ^ { \circ }$
C) $96 ^ { \circ }$
D) $43 ^ { \circ }$
E) $68 ^ { \circ }$

Question 63

Find the values of $\sin \theta$ and $\tan \theta$ If the point $( - 3,4 )$ Is on the terminal side of $\theta$

A) $\sin \theta = \frac { 3 } { 5 }$
, $\tan \theta = - \frac { 4 } { 3 }$

B) $\sin \theta = \frac { 4 } { 5 }$
, $\tan \theta = - \frac { 3 } { 4 }$

C) $\sin \theta = - \frac { 3 } { 5 }$
, $\tan \theta = - \frac { 4 } { 3 }$

D) $\sin \theta = \frac { 4 } { 5 }$
, $\tan \theta = - \frac { 4 } { 3 }$

E) $\sin \theta = \frac { 4 } { 5 }$
, $\tan \theta = - \frac { 5 } { 3 }$

Question 64

A $200$ ft guy wire is attached to the top of a tower. If the wire makes a $55 ^ { \circ }$
Angle with the ground, how tall is the tower?

A) $163.8$ ft
B) $90 \mathrm { ft }$
C) $265 \mathrm { ft }$
D) $85.4 \mathrm { ft }$
E) $255.6 \mathrm { ft }$

Question 65

Find the degree measure that corresponds to each radian measure. ,

A)
B)
C)
D)
E)

Question 66

To find the distance across a river a surveyor chooses points $A$ and $B$ , $50$ m apart and a reference point $C$ on the opposite side of theriver. Approximate the distance, $b$ , between $A$ and $C$ given that $\angle \mathrm { A } =54 ^ { \circ }$ and $\angle B = 77 ^ { \circ }$ .

Question 67

Use Heron's formula to find the area of the triangle with sides of length 12, 18, and 24

A)
B)
C)
D)
E) none of these

Question 68

A plane leaves the airport at 4:00 p.m. and heads in the direction N $10 ^ { \circ }$ E at a constant speed of $250$ mi/h. Forty-five minutes later, a jet leaves the same airport and flies N $60 ^ { \circ }$ W at a constant speed of $450$ mi/h. Find the distance between the plane and the jet at 5:30 p.m

A) $409.8 \mathrm { mi }$
B) $904.8 \mathrm { mi }$
C) $309 \mathrm { mi }$
D) $7.5 \mathrm { mi }$
E) $77 \mathrm { mi }$

Question 69

Find the length of an arc of a circle of radius $3$ m if the arc subtends a central angle of $75 ^ { \circ }$

A) $s = \frac { 5 \pi } { 4 } \mathrm { ft }$

B) $s = 25 \pi \mathrm { ft }$

C) $s = \frac { 3 \pi } { 8 } \mathrm { ft }$

D) $s = \frac { 9 \pi } { 4 } \mathrm { ft }$

E) $s = 5 \mathrm { ft }$

Question 70

Find the exact value of the expression. $\tan \left( \sin ^ { - 1 } \frac { 3 } { 5 } \right)$

A) $3 / 5$
B) $5 / 3$
C) $- 3 / 5$
D) $- 5 / 3$
E)none of these

Question 71

Find the radian measure that corresponds to each degree measure. ,

A)
B)
C)
D)
E)

Question 72

How many revolutions will a bicycle wheel of diameter $25$ inches make as the bicycle travels a distance of $3$ Miles?

A) $2,193$
B) $1,760$
C) $2,421$
D) $3,806$
E) $5,280$

Question 73

To find the distance across a river a surveyor chooses points $A$ and $B$ , $50$ M apart and a reference point $C$ On the opposite side of the river. Approximate the distance, $b$ , between $A$ And $C$ Given that $\angle \mathrm { A } = 5 4^ { \circ }$ And $\angle B = 77 ^ { \circ }$

A) $54.6 \mathrm {~m}$
B) $77.6 \mathrm {~m}$
C) $64.6 \mathrm {~m}$
D) $45.6 \mathrm {~m}$
E) $85.6 \mathrm {~m}$

Question 74

From the top of a $250$ ft lighthouse, the angle of depression to a ship in the ocean is $18 ^ { \circ }$ ) How far is the ship from the base of the lighthouse?

A) $269 \mathrm { ft }$
B) $769 \mathrm { ft }$
C) $810 \mathrm { ft }$
D) $81.4 \mathrm { ft }$
E) $567.3 \mathrm { ft }$

Question 75

A 76-ft tree casts a shadow that is 100 ft long. What is the angle of elevation of the sun?

A)
B)
C)
D)
E)none of these

Question 76

Find the exact value of the expression. $\left( \sin 60 ^ { \circ } \right) ^ { 2 } - \left( \cos 60 ^ { \circ } \right) ^ { 2 }$

A) $\frac { 3 } { 4 }$

B) $\frac { 1 } { 2 }$

C) $\frac { \sqrt { 3 } } { 4 }$

D) $\frac { 3 } { 2 }$

E) $\frac { 1 } { 4 }$

Question 77

If for
In quadrant I, find

A)
B)
C)
D)
E)

Question 78

Find $\sin \theta$ and $\cot \theta$
Given that $\sec \theta = - 2$ And $\tan \theta > 0$

A) $\sin \theta = - \frac { 3 . \sqrt { 5 } } { 2 }$
, $\cot \theta = \frac { 2 - \sqrt { 9 } } { 3 }$

B) $\sin \theta = \frac { \sqrt { 5 } } { 2 }$
, $\cot \theta = - 2$

C) $\sin \theta = - \frac { \sqrt { 5 } } { 2 }$
,
$\cot \theta=-\frac{\sqrt{3}}{3}$

D) $\sin \theta = - \frac { 1 } { 2 }$
, $\cot \theta=\frac{\sqrt{3}}{3}$

E) $\sin \theta=-\frac{\sqrt{3}}{2}$
, $\cot \theta=\frac{\sqrt{3}}{3}$

Question 79

Solve the triangle.

A)
B)
C)
D)
E) none of these

Question 80

Find the acute angle that is formed by the line and the x-axis

A)
B)
C)
D)
E)