11/12

MATH 152

COLLEGE ALGEBRA AND TRIGONOMETRY

TEST 2 REVIEW

TO THE STUDENT:

To best prepare for Test 2, do all the problems on separate paper.

The pages referenced are in the textbook and the answers to odd-numbered problems are

given in the back of the book.

The answers to the even-numbered problems and to the “Additional Problems” problems are

included at the end of this Review Sheet.

PART 1 – NON CALCULATOR

DIRECTIONS:

Read the Chapter 3 Review, pages 165 – 170.

The problems on this part of the Review Sheet are similar to those you can expect on the

non-calculator part of the test. For that reason, you should do these problems without your

graphing calculator.

Show all your steps.

Support all answers with appropriate reasoning.

Use graph paper for graphs unless the problem asks you for a sketch.

Label all graphs completely

Answer application problems using complete sentences.

If a table is used to support an answer, include the relevant rows.

A. CHAPTER 3 and 4 REVIEW

1. (page 170: 1) Convert to radian measure in terms of .

a) 60° b) 45° c) 90°

2. (page 170: 2) Convert to degree measure.

a) 6

b) 2

c) 4

3. (page 170: 8) Find the value of sin and tan if the terminal side of contains P(–4, 3).

4. (page 170: 9) Is it possible to find a real number x such that sin x is negative and csc x is

positive? Explain.

5. (page 171: 17) List all angles that are conterminal with 6

rad, 3 3 .

Explain how you arrived at your answer.

6. (page 171: 24) If sin sin , , are angles and necessarily conterminal

Explain.

Solution #1, 2

Solution #3, 4, and 5

2

7. (page 171: 25) From the following display on a graphing calculator, explain how you

would find csc x without finding x. Then find csc x to four decimal places.

8. (page 171: 26) Find the tangent of 0, 2

, , and 3

2.

From problems 9-12 , find the exact value of each without using a calculator.

9. (page 171: 35) 7

cot4

10. (page 171: 37) 3

cos2

11. (page 171: 39) 4

3sec

12. (page 171: 41) cot 3

13. (page 171: 49) Find the least positive exact value of in radian measure such that

1

sin2

.

14. (page 171: 50) Find the exact value of each of the other five trigonometric functions if

2

sin tan 05

and

15. (page 171: 60) One of the following is not an identity. Indicate which one.

a) 1

cscsin

xx

b) 1

cottan

xx

c) sin

tancos

xx

x

d) 1

secsin

xx

e) 2 2sin cos 1x x f) cos

cotsin

xx

x

16. (page 171: 64) An angle in standard position intercepts an arc of length 1.3 units on a

unit circle with center at the origin. Explain why the radian measure of the angle is also

1.3.

17. (page 171: 65) Which circular functions are not defined for x = k , k any integer?

Explain.

For problems 18-19, do the functions appear to be periodic with period less than 4?

18. (page 138: 6)

Solution 9 - 11

Solution 12 - 14

3

19. (page 138: 8)

20. (page 139: 20) For the graph below, describe your height, h = f (t), above the ground on

different ferris wheels, where h is in meters and t is time in minutes. You boarded the

wheel before t = 0. Determine the following: your position and direction at t = 0, how

long it takes the wheel to complete one full revolution, the diameter of the wheel, at what

height above the ground you board the wheel, and the length of time the graph shows you

riding the wheel. The boarding platform is level with the bottom of the wheel.

21. (page 214: 2) For 6sin 4y t , state the period, amplitude, and midline.

22. Based on the graphs below, find the formula for the trigonometric

function.

a) (page 214: 14)

b) (page 214: 20)

23. (page 214: 25) Find a formula, using the sine function, for your height above ground

after t minutes on the Ferris wheel, Graph the function to check that is correct.

A ferris wheel is 20 meters in diameter and boarded in the six o'clock position from a

platform that is 4 meters above the ground. The wheel completes one full revolution

every 2 minutes. At t = 0 you are in the twelve o'clock position.

solution #20

Solution #22a)

Solution #22b)

Solution #23

4

B. CHECK YOUR UNDERSTANDING

1. Is sin( ) sinx x for all values of x? Give an explanation for your answer.

Are the statements in Problems 2 – 10 true or false? Give an explanation for your

answer.

2. The function sin( )x has period .

3. An angle of one radian is about equal to an angle of one degree.

4. The amplitude of 3sin(2 ) 4y x is 3 .

5. The amplitude of 25 10cosy x is 25.

6. The period of 25 10cosy x is 2 .

7. The maximum y-value of 25 10cosy x is 10.

8. The minimum y-value of 25 10cosy x is 15.

9. The midline equation for 25 10cosy x is 35y .

10. The function ( ) cos(3 )f x x has a period three times a large as the function

( ) cosg x x .

C. ADDITIONAL PROBLEMS:

1. For a–b, use the figure below

N

p

m

θ

P n M

a) State each of the six trigonometric ratios of θ in terms of m, n, and p.

b) Solve the triangle given that 30 and 3m

2. Give an example of two coterminal angles with their terminal sides in quadrant 3. Give

answers in degrees and radians.

3. Write expressions for the six trigonometric functions of the angle θ when θ is an angle is

standard position and ( , )P a b is a point on the terminal side of θ. Let r be the distance

from (0,0) to P.

4. Give three examples of quadrantal angles in radians.

Solution #1

5

5. Write the definition of periodic function.

6. Use periodic properties to find the value of:

a) 7

sin3

b) 25

cos6

c) 9

sin4

7. For Figures 1 and 2 below, give the amplitude, midline, and period.

Figure 1 Figure 2

8. Sketch the graph of each of the six trigonometric functions on the x-interval [ 2 ,2 ] .

Label intercepts and asymptotes. State the domain, range, and period of each function.

9. Solve each equation; in each case, write the complete solution set.

a) 2

cos2

x b) 1

sin2

x

c) tan 1x d) sin 2x

e) csc 2x f) tan 3x

10. Suppose that an equation of the form sin x c has the solutions x a and x b in the

interval 0,2 . Write the general solutions to this equation.

11. Identify the basic function, state the transformations, and sketch the graph. Label all the

important points and features on the graph.

a) 3y x b) 1

32

yx

12. Sinusoidal functions may be written in the form:

( ) sin( ( ))f x A B x h k or ( ) cos( ( ))g x A B x h k

In terms of A, B, h, and k, give the:

x

y

X

Y

Solution #7

Solution #9a)-9c)

Solution #9d) - 9f)

Solution #11

Solution #6

6

a) Midline

b) Amplitude

c) Period

d) Horizontal shift

For problems 13 – 18:

For the primary cycle of each of the functions given, identify:

Vertical Shift (if any)

Horizontal Shift (if any)

Reflection (if any)

Midline

Amplitude

Maximum Value

Minimum Value

Period

Beginning

Quarter Distance

First Quarter Point

Midpoint

Third Quarter Point

End

Without using your graphing calculator, graph one complete cycle of the function. Label

all the important points and lines of the graph.

13. 4 siny x 14. cos4

y x

15. cos3

y x 16. cos 212

y x

17. 2siny x 18. 1

cos2 6

y x

For problems 19 – 26: Solve each equation algebraically. Give the general solution.

19. 1

sin(2 )2

20. 2cos( ) 3x

21. 3 csc(2 ) 2x 22. 2cos(3 ) 3x

23. tan(2 ) 1x 24. sec(5 ) 2x

25. 4 2cos(3 ) 0 26. 4 8sin(3 ) 0t

27. Express each of the following descriptions using an appropriate function. Be sure to

define your variables.

a) The cost of renting a car for one day is $40 plus $0.35 per mile.

Solution #13 Solution #14

Solution #15 Solution #16

Solution #17 Solution #18

Solution #19, 20

Solution #21, 22

Solution #23, 24

solution #25, 26

7

b) The population of rabbits starts at 400, increases to 450, decreases back to 400, then

down to 350, then increases back to 400, all over the course of 5 years.

c) A town’s population was 1100 in 1990 and 3 years later had declined at a constant

rate to 500.

28. The following equations give animal populations as functions of time, t, in years since the

year 2000. Describe the growth of each population in words.

a) 800 12P t

b) 800 15P t

c) 2

150sin 8003

P t

29. Suppose 2

2

4 12( )

4

x xy g x

x

a) What is the domain?

b) Find (0)g

c) Find all values of x for which ( ) 0g x .

d) What are the x-intercepts?

e) What are the y-intercepts?

30. A T-Shirt printing company charges a set-up fee of $10 for each order, plus the cost

per shirt shown below.

Number of shirts, n Cost per shirt in $,

C

0–10 10

11–20 9

21–30 8

> 30 7

Express C, the total cost in dollars, as a piecewise function of n, the number of shirts

ordered.

31. If 2( ) 2f x x x , find the average rate of change between

(x, f (x)) and (x + h, f (x + h)).

32. On graph paper, graph the following piecewise function and state the domain and range.

2

6, 2

( ) , 2 0

sin , 0

x x

f x x x

x x

Solution #29

Solution #30, 31

8

33. Let 2( ) 5 3h x x x

Find and simplify:

a) (2)h b) ( )h t c) ( 2)h x d) 2 ( ) 2h x

34. Suppose you are on a Ferris wheel (that turns in a counter–clockwise direction) and that

your height, in meters, above the ground at time, t, in minutes is given by

( ) 15sin 152

h t t

a) How high above the ground are you at time 0t ?

b) At what time, t, will you be at the maximum height?

c) What is the radius of the wheel?

d) How long does one revolution take?

PART 2 –CALCULATOR

DIRECTIONS:

You may use your graphing calculator on this part of the review sheet.

Support all answers with appropriate reasoning.

If a graph is used to support an answer, include a sketch.

If a table is used to support an answer, include the relevant rows.

Show all your steps.

Answer application problems using complete sentences.

A. TEXT

1. (page 140: 32 a, b, c, e, g) Use a calculator or a computer to decide whether eac of the

following functions is periodic or not.

a) ( ) sinx

f x b) ( ) sinf xx

c) ( ) sinf x x x

d) ( ) sinf x x e) ( ) sinf x x

2. (page 172: 74) An alternating current generator produces an electrical current (measured

in amperes) that is described by the equation 30sin 120 60I t where t is time in

seconds. What is the current I when t = 0.015 sec? (Give answers to one decimal place)

B. CHECK YOUR UNDERSTANDING

Are the statements in Problems 1 – 2 true for all values of x? Give an explanation for your

answer.

Solution #33, 34

9

1. 1 cos1

coscosx x

2. cos( 1) cos cos1x x

C. ADDITIONAL PROBLEMS

In problems 1 – 6, evaluate each expression to 2 decimal places.

1. csc15

2. 2

sec5

3. 4

cot

4. sin30

5. 2

cos

6. 8

sec3

7. Suppose 2

20( )

4

xf x

x.

a) What is the domain of ( )f x ? Explain how you know.

b) Use your calculator to graph the function. You will need to determine a suitable

window. Draw your graph on graph paper and label your scale.

c) What is the range of this function?

d) From your graph, give the approximate intervals where the function is increasing and

where it is decreasing.

e) From your graph, give the approximate intervals where the graph is concave up and

where it is concave down.

8. Consider the equation 2 cos2 1x .

a) Find the exact solutions in the interval 0 2x .

b) Verify the solutions by solving the equation graphically: Graph two functions (one for

each side of the equation) and find the intersections. Write down the solutions to two

decimal places and verify that they match your solutions from part a) above.

Solution #7

Solution #8a)

Solution #8b)

11/12

MATH 152 Test 2 Review Answers

PART 1 – NON CALCULATOR

A. CHAPTER 3 REVIEW

1. a) 3

b) 4

c) 2

2. a) 30 b) 90 c) 45

3. 3 3

sin , tan5 4

4. No, since 1

cscsin

xx

, when one is positive so

is the other.

5. 11 13

,6 6

. When the terminal side of the angle

is rotated any mulitiple of a complete revolution

(2 rad) in either directions, the resulitng angle

will be coterminal with the original. In this case,

for the restricted interval, this happens for

26

.

6. No, angles and are not necessarily

coterminal. The terminals sides of and

contain points with opposite x- coordinates and

the same y-coordinate.

7. Use the reciprocal identity:

1 1

csc 1.1636sin 0.8594

xx

8. tan 0 0, tan is undefined, tan 02

and 3

tan is undefined2

.

9. –1

10. 0

11. –2

12. not defined

13. 7

6

14. 21 2

cos , tan5 21

5 5 21

csc , sec , cot2 221

15. d) is not an identity.

16. based on definition of radian of an angel. s

r.

When r = 1, then 1.31

s s

rrad.

17. functions are not defined for x = k , only for

sin (x) in the denominator for any integer k.

That means 1 cos

csc cotsin sin

xx and x

x x

So, csc x and cot x are not defined for x = k ,

18. Not periodic

19. Not periodic

20. At t = 0, the at 20 meters above the ground and

ascending . It takes the wheel 5 minutes to

complete a full rotation. The diameter of the

wheel is 40 meters. The minimum of the

function is h = 0 so you board and get off at

ground level. The function completes 2.25

periods, so you ride the wheel 5 (2.25)= 11.25

minutes

21. Amplitude: 6

Midline: y = 0

Period: 2

22. a) 1

( ) 2sin 22

g t t

b) 2cos2

xy

23. ( ) 14 10sin2

f t t

B. Check your understanding

1. TRUE. sin( ) sinx x for all x

The sine function is an odd function

2. FALSE. The period of sin( )y x , is

2

P = 2

3. FALSE. 1 57.3radian

4. FALSE. The amplitude of 3sin(2 ) 4y x

is 3.

5. FALSE. The amplitude of 25 10cosy x

is 10.

6. TRUE. The period of the function

25 10cosy x is 2

21

, the same as the

period of the function cosy x .

7. FALSE. The maximum y-value of

25 10cosy x is 25+10= 35.

8. TRUE. The minimum y-value of

25 10cosy x is 25−10=15.

9. FALSE. The equation of the midline for

25 10cosy x is 25y .

11

10. FALSE. The period of the function

( ) cos(3 )f x x is 2

3 and the period of

( ) cosg x x is 2 . The period of f(x) is 1/3

that of g(x).

C. Additional Problems

1. a)

sin , cos , tan

csc , sec , cot

m n mp p n

p p nm n m

b) 60 , 6, 3 3N p n

2. Answers vary. A good answer could be

4 10

(240 ) and (600 ).33

3. sin , cos , tan , 0b a b

ar r a

csc , 0, sec , 0

cot , 0

r rb a

b a

ab

b

4. Answers could be

3

0, , ,2 or any multiple of2 2

5. A function f is a periodic function if there is a

positive number p such that f(x+p) = f (x) for all

x in the domain of f.

6. a) 7 3

sin sin3 3 2

b) 325

cos cos6 6 2

c) 9 1

sin sin4 4 2

7. Figure 1 : Amplitude = 1,

Midline: y = 1 , Period = 2 .

Figure 2: Amplitude = 2,

Midline: y = −1 , Period = 2.

8. siny x

Domain : ( , )

Range: 1,1

Period: 2

cos( )y x

Domain : ( , )

Range: 1,1

Period: 2

tan( )y x

Domain : Set of all real numbers R except

2

k , with k an integer.

Range: ( , )

Period:

csc( )y x

Domain: All real numbers x, except

x k , k an integer.

Range: ( , 1] [1, )

Period : 2

sec( )y x

x

y

x

y

x

y

12

x

y

(0, 3)

Domain: All real numbers x, except

,2

x k , k an integer.

Range: ( , 1] [1, )

Period: 2

cot( )y x

Domain: All real numbers x, except

x k , k an integer.

Range: ( , )

Period :

9. a) 7

24

2 ,4

kx k x

b) 7 11

2 , 26 6

x k x k

c) 4

x k

d) No solution

e) 7 11

2 , 26 6

x k x k

f) 3

x k

10. The general solutions are

2 , 2x a k x b k

11. a) Basic function: y x

Reflection across the y-axis

Reflection across the x-axis

Vertical Shift 3 units up

b) Basic function :1

xy

Horizontal shift 2 units right

Vertical shift 3 units up

12. a) Midline: y k

b) Amplitude: A

c) 2

PB

d) Horizontal shift: h units. If h > 0, shift

right. If h < 0, shift left.

13. Reflection across the x-axis

Vertical shift 4 units downa

Midline: y = −4

Amplitude: 1

Maximum Value: y = −3

Minimum Value: y = −5

Period: 2

Beginning: 0, 4

Quarter Distance: 2

First Quarter Point: , 52

Midpoint: , 4

x

y

(2, 3)

13

Third Quarter Point: 3

, 32

End: 2 , 4

14. Horizontal shift 4

units left

Midline: y = 0

Amplitude: 1

Maximum Value: y = 1

Minimum Value: y = −1

Period: 2

Beginning: ,14

Quarter Distance: 2

First Quarter Point: ,04

Midpoint: 3

, 14

Third Quarter Point: 5

,04

End: 7

,14

15. Horizontal shift 3

units left

Reflection across the x-axis

Midline: y = 0

Amplitude: 1

Maximum Value: y = 1

Minimum Value: y = −1

Period: 2

Beginning: , 13

Quarter Distance: 2

First Quarter Point: ,06

Midpoint: 2

,13

Third Quarter Point: 7

,06

End: 5

, 13

16. Horizontal shift 12

units right

Vertical shift 2 units up

Midline: y =2

Amplitude: 1

Maximum Value: y = 3

Minimum Value: y =1

Period: 2

Beginning: ,312

Quarter Distance: 2

First Quarter Point: 7

, 212

Midpoint: 13

,112

Third Quarter Point: 19

,212

End: 25

,312

4

3

4

5

47

44

-1

1

x

y

cos3

y x

x

y4 siny x

14

17. Vertical stretch and Reflection across the x-axis

Midline: y = 0

Amplitude: 2

Maximum Value: y = 2

Minimum Value: y = −2

Period: 2

Beginning: 0,0

Quarter Distance: 2

First Quarter Point: , 22

Midpoint: ,0

Third Quarter Point: 3

, 22

End: 2 , 0

18. Horizontal shift 3

units right

Horizontal stretch

Midline: y = 0

Amplitude: 1

Maximum Value: y = 1

Minimum Value: y = −1

Period: 4

Beginning: ,13

Quarter Distance:

First Quarter Point: 4

,03

Midpoint: 7

, 13

Third Quarter Point: 10

,03

End: 13

,13

19. 5

,12 12

k k

20. 11

2 , 26 6

x k x k

21. ,6 3

x k x k

22. 2 11 2

,18 3 18 3

x k x k

23. 8 2

kx

24. 2 7 2

,20 5 20 5

k kx x

25. No solution.

26. 7 2 11 2

,18 3 18 3

k kt t

27. a) ( ) 40 0.35C m m , m , miles, C(m), cost

b) 2

( ) 50sin 4005

tP t , t, time and P(t)

population.

c) ( ) 1100 200P x x , P(x) is the town

population and x is the number of years

since 1990.

28. a) The animal population started at 800 and

decreased at an average rate of 12 animals

per year.

x

y2siny x

-1

1

2

3

4

12

7

12

13

12

19

12

25

12

,312

7,2

12 13 ,112

19 ,212

25 ,312

3

4

3

7

3

10

3

13

3

,13

7 , 13

13 ,13

15

b) The animal population started at 800 and

increased at an average rate of 15 animals

per year.

c) The animal population started starts at 800,

then increases to 950, decreases back to 800,

then down to 650, then increases back

to 800, all over the course of 3 years.

29. a) ( , 2) ( 2,2) (2, )

b) (0) 3g

c) 6x

d) ( 6,0)

e) (0,3)

30. C(n) =

10 10 , 0 10

10 9 , 11 20

10 8 , 21 30

10 7 , 30

n if n

n if n

n if n

n if n

31.

32. Domain: all real numbers;

Range: [ 4, )

33. a) (2) 11h

b) 2( ) 5 3h t t t

c) 2( 2) 9 11h x x x

d) 22 ( ) 2 2 10 8h x x x

34. a) 15 meters above the ground

b) The maximum height happens at

t = 1 minute.

c) The radius of the wheel is 15 meters

d) One revolution takes 4 minutes.

PART 2 – CALCULATOR

A. Text

1. a) Periodic

b) Not periodic

c) Not periodic

d) Periodic

e) Periodic

2. −17.6 amperes

B. Check your Understanding

1. FALSE. 1 cos1

coscosx x

, for some x.

Example: x = 1, cos(1) 1

2. FALSE. cos( 1) cos cos 1x x for

some x. Example: x = 0, cos(1) 1 cos(1)

C. Answers to Additional Problems 1. 4.81

2. 3.24

3. 0.31

4. −0.99

5. 0.80

6. −2

7. a) The domain is ( , ) . The function is

defined for all Real numbers because its

denominator never equals zero.

b)

Window: 30,30 by 5,5

c) 5,5

d) Increasing: ( 2,2)

Decreasing: ( , 2) (2, )

e) Concave up:

Approximately ( 4,0) (4, )

Concave down:

Approximately ( , 4) (0,4)

8. a) 3 5 11 13

, , ,8 8 8 8

x

b) 1.18, 1.96, 4.32, 5.11x

(–2, –4)

x

y

2 cos2y x

y = –1

( ) ( )4 2

f x h f xx h

h