Download math 152 – college algebra and trigonometry

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)
.
Solution #1, 2
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.
Solution #3, 4, and 5
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
Explain.
sin ,
, are angles
and
necessarily conterminal
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) cot
7
4
10. (page 171: 37) cos
3
2
11. (page 171: 39) sec
Solution 9 - 11
4
3
12. (page 171: 41) cot 3
Solution 12 - 14
13. (page 171: 49) Find the least positive exact value of
1
.
sin
2
in radian measure such that
14. (page 171: 50) Find the exact value of each of the other five trigonometric functions if
2
sin
and tan
0
5
15. (page 171: 60) One of the following is not an identity. Indicate which one.
a) csc x
1
sin x
b) cot x
d)
1
sin x
e)
sec x
1
tan x
c)
tan x
sin x
cos x
sin2 x cos2 x 1
f)
cot x
cos x
sin 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)
2
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.
solution #20
21. (page 214: 2) For y
6sin t 4 , state the period, amplitude, and midline.
22. Based on the graphs below, find the formula for the trigonometric
function.
a) (page 214: 14)
Solution #22a)
b) (page 214: 20)
Solution #22b)
Solution #23
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.
3
B. CHECK YOUR UNDERSTANDING
1. Is sin( x)
sin 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 y
5. The amplitude of y
6. The period of y
3sin(2x) 4 is 3 .
25 10cos x is 25.
25 10cos x is 2 .
7. The maximum y-value of y
25 10cos x is 10.
8. The minimum y-value of y
25 10cos x is 15.
9. The midline equation for y
25 10cos x is y 35 .
10. The function f ( x) cos(3x) has a period three times a large as the function
g ( x) cos x .
C. ADDITIONAL PROBLEMS:
1. For a–b, use the figure below
N
Solution #1
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 m 3
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.
4
5. Write the definition of periodic function.
6. Use periodic properties to find the value of:
a) sin
7
3
b) cos
25
6
c) sin
Solution #6
9
4
7. For Figures 1 and 2 below, give the amplitude, midline, and period.
y
Y
X
x
Figure 1
Figure 2
Solution #7
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) cos x
2
2
c) tan x 1
e) csc x
2
b) sin x
1
2
d) sin x
2
f) tan x
3
Solution #9a)-9c)
Solution #9d) - 9f)
10. Suppose that an equation of the form sin x c has the solutions x
interval 0, 2 . Write the general solutions to this equation.
a and x b in the
11. Identify the basic function, state the transformations, and sketch the graph. Label all the
important points and features on the graph.
a) y 3
x
b) y
1
x 2
3
12. Sinusoidal functions may be written in the form:
f ( x) Asin( B( x h)) k or g ( x) A cos( B( x h)) k
In terms of A, B, h, and k, give the:
5
Solution #11
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. y
4 sin x
15. y
cos x
17. y
2sin x
Solution #13
3
Solution #15
14. y
cos x
16. y
cos x
18. y
cos
Solution #17
Solution #14
4
12
1
x
2
6
2
Solution #16
Solution #18
For problems 19 – 26: Solve each equation algebraically. Give the general solution.
1
19. sin(2 )
20. 2cos( x)
3
Solution #19, 20
2
21.
3 csc(2 x) 2
22. 2cos(3x)
3
23. tan(2 x) 1
24. sec(5x)
25. 4 2cos(3 ) 0
26. 4 8sin(3t ) 0
2
Solution #21, 22
Solution #23, 24
solution #25, 26
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.
6
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) P 800 12t
b) P 800 15t
c) P 150sin
29. Suppose y
2
t
3
g ( x)
800
x 2 4 x 12
x2 4
a) What is the domain?
Solution #29
b) Find g (0)
c) Find all values of x for which g ( x) 0 .
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.
2 x 2 x , find the average rate of change between
31. If f ( x )
(x, f (x)) and (x + h, f (x + h)).
Solution #30, 31
32. On graph paper, graph the following piecewise function and state the domain and range.
x 6, x
f ( x)
x2 ,
sin x,
2
2
x
x 0
0
7
x2 5x 3
33. Let h( x)
Find and simplify:
a) h(2)
b) h(t )
c) h( x 2)
d) 2h( x) 2
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
h(t ) 15sin
2
t
15
a) How high above the ground are you at time t
0?
Solution #33, 34
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)
f ( x ) sin
d) f ( x)
x
sin x
b) f ( x ) sin
e)
f ( x)
x
c)
f ( x)
x sin x
sin x
2. (page 172: 74) An alternating current generator produces an electrical current (measured
in amperes) that is described by the equation I 30sin 120 t 60 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.
8
1. cos
1
x
cos1
cos x
2. cos( x 1) cos x cos1
C. ADDITIONAL PROBLEMS
In problems 1 – 6, evaluate each expression to 2 decimal places.
1. csc
2. sec
3. cot
15
2
5
4
4. sin 30
5. cos
6. sec
2
8
3
7. Suppose f ( x)
20 x
.
x2 4
Solution #7
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 cos 2 x
1.
a) Find the exact solutions in the interval 0
x 2 .
Solution #8a)
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 #8b)
9
11/12
MATH 152 Test 2 Review Answers
PART 1 – NON CALCULATOR
1
cos x
and cot x
sin x
sin x
So, csc x and cot x are not defined for x = k ,
That means csc x
A. CHAPTER 3 REVIEW
1. a)
b)
3
2. a) 30
c)
4
b) 90
sin
4.
No, since csc x
5.
6.
7.
8.
c) 45
3
, tan
5
3.
2
3
4
1
, when one is positive so
sin x
is the other.
11 13
. When the terminal side of the angle
,
6
6
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
2 .
6
No, angles and
are not necessarily
coterminal. The terminals sides of and
contain points with opposite x- coordinates and
the same y-coordinate.
Use the reciprocal identity:
1
1
csc x
1.1636
sin x 0.8594
tan 0
0, tan
and tan
3
2
2
is undefined, tan
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
1
22. a) g (t )
2sin
t
2
2
x
b) y 2cos
2
23. f (t ) 14 10sin
t
2
B. Check your understanding
1.
TRUE. sin( x)
sin x for all x
The sine function is an odd function
2.
FALSE. The period of y
P
0
2
sin( x ) , is
=2
3.
FALSE. 1 radian
57.3
4.
FALSE. The amplitude of y
is 3.
5.
FALSE. The amplitude of y
is 10.
6.
TRUE. The period of the function
2
2 , the same as the
y 25 10cos x is
1
period of the function y cos x .
7.
FALSE. The maximum y-value of
y 25 10 cos x is 25+10= 35.
8.
TRUE. The minimum y-value of
y 25 10 cos x is 25−10=15.
9.
FALSE. The equation of the midline for
y 25 10cos x is y 25 .
is undefined .
–1
0
–2
not defined
7
13.
6
21
14. cos
, tan
5
9.
10.
11.
12.
2
21
5
5
csc
, sec
, cot
2
21
15. d) is not an identity.
21
2
16. based on definition of radian of an angel.
s s
1.3 rad.
r 1
17. functions are not defined for x = k , only for
sin (x) in the denominator for any integer k.
When r = 1, then
s
.
r
3sin(2 x ) 4
25 10 cos x
10. FALSE. The period of the function
2
and the period of
f ( x) cos(3x) is
3
g ( x) cos x is 2 . The period of f(x) is 1/3
that of g(x).
y cos( x )
Domain : (
Range:
, )
1,1
Period: 2
y
C. Additional Problems
m
, cos
p
p
, sec
m
sin
1.
a)
csc
2.
3.
4.
5.
6.
7.
8.
n
, tan
p
p
, cot
n
m
n
n
m
x
b)
N 60 , p 6, n 3 3
Answers vary. A good answer could be
4
10
(240 ) and
(600 ).
3
3
b
a
b
sin
, cos
, tan
,a 0
r
r
a
r
r
csc
, b 0, sec
,a 0
b
a
a
cot
,b 0
b
Answers could be
3
0, , ,2 or any multiple of
2
2
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.
7
3
sin
a) sin
3
3
2
b)
cos
25
6
c)
sin
9
4
cos
sin
4
y tan( x)
Domain : Set of all real numbers R except
k , with k an integer.
2
Range: ( , )
Period:
y
x
3
6
2
1
2
y csc( x )
Domain: All real numbers x, except
x k , k an integer.
Range: ( , 1] [1, )
Period : 2
Figure 1 : Amplitude = 1,
Midline: y = 1 , Period = 2 .
Figure 2: Amplitude = 2,
Midline: y = −1 , Period = 2.
y sin x
Domain : ( , )
Range:
1,1
Period: 2
y
x
y sec( x )
11
Domain: All real numbers x, except
x
Range: (
Period: 2
Vertical Shift 3 units up
k , , k an integer.
2
, 1] [1, )
y
(0, 3)
x
b) Basic function : y
1
x
Horizontal shift 2 units right
Vertical shift 3 units up
y cot( x )
Domain: All real numbers x, except
x k , k an integer.
Range: ( , )
Period :
y
(2, 3)
x
12. a) Midline: y
k
b) Amplitude: A
2
B
d) Horizontal shift: h units. If h > 0, shift
right. If h < 0, shift left.
c)
9.
a)
x
b)
x
4
7
6
2 k, x
2 k, x
k
4
d) No solution
7
e) x 6 2 k , x
c)
f)
7
4
2 k
11
6
2 k
11
6
2 k
P
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
x
x
k
3
10. The general solutions are
x a 2k , x b 2k
Quarter Distance:
2
First Quarter Point:
11. a) Basic function: y
x
Reflection across the y-axis
Reflection across the x-axis
Midpoint:
12
, 4
2
, 5
Third Quarter Point:
Reflection across the x-axis
Midline: y = 0
Amplitude: 1
Maximum Value: y = 1
Minimum Value: y = −1
Period: 2
3
, 3
2
End: 2 , 4
y
y
4 sin x
x
Beginning:
3
, 1
Quarter Distance:
2
First Quarter Point:
Third Quarter Point:
4
4
1
2
y
4
cos x
-1
,0
3
, 1
4
16. Horizontal shift
Third Quarter Point:
7
y
x
First Quarter Point:
End:
, 1
,1
Quarter Distance:
Midpoint:
,1
Beginning:
units right
12
12
,3
Quarter Distance:
2
First Quarter Point:
4
3
4
5
4
7
4
Midpoint:
3
units left
End:
13
7
,2
12
13
,1
12
Third Quarter Point:
15. Horizontal shift
3
Vertical shift 2 units up
Midline: y =2
Amplitude: 1
Maximum Value: y = 3
Minimum Value: y =1
Period: 2
5
,0
4
4
4
7
,0
6
3
Midline: y = 0
Amplitude: 1
Maximum Value: y = 1
Minimum Value: y = −1
Period: 2
Beginning:
5
End:
units left
,0
2
,1
3
Midpoint:
14. Horizontal shift
6
25
,3
12
19
,2
12
First Quarter Point:
4
3
2
1
12
,3
25
7
,2
12
13
-1 12
19
12
13
12
7
12
12
12
,3
13
End:
3
2
, 2
19.
21. x
23.
x
24.
25.
26.
27.
units right
3
6
k,
x
2
k,
18 3
k
x
8
2
2 k
x
,
20
5
No solution.
7
2 k
t
,
18
3
a) C(m) 40
b)
Horizontal stretch
Midline: y = 0
Amplitude: 1
Maximum Value: y = 1
Minimum Value: y = −1
Period: 4
Beginning:
2 k,
22. x
2 sin x
3
6
c)
P(t )
3
5
12
11
x
6
k,
12
20. x
3
,2
2
End: 2 , 0
18. Horizontal shift
7
3
4
3
2
Third Quarter Point:
y
13
,1
3
,0
y
,1
3
25
12
7
First Quarter Point:
Midpoint:
10
,0
3
Third Quarter Point:
,2
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:
7
, 1
3
Midpoint:
,1
19
12
4
,0
3
10
3
3
,1
13
3
, 1
k
2 k
k
3
x
11
18
x
7
20
2
k
3
2 k
5
11
2 k
18
3
0.35m , m , miles, C(m), cost
t
50sin
2 t
5
400 , t, time and P(t)
population.
P( x) 1100 200 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.
,1
Quarter Distance:
14
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)
c)
d)
d) Periodic
e) Periodic
2. −17.6 amperes
B. Check your Understanding
1.
10 10n, if 0
30. C(n) =
n 10
10 9n, if 11 n
20
10 8n, if 21 n 30
10 7n, if n
30
f ( x h) f ( x )
4 x 2h
h
32. Domain: all real numbers;
Range: [ 4, )
31.
Window:
(–2, –4)
b)
c)
d)
h(2)
11
h(t )
2
t
h( x 2)
30,30 by
5,5
5,5
c)
33. a)
cos1
, for some x.
x
cos x
Example: x = 1, cos(1) 1
2. FALSE. cos( x 1) cos x cos 1 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)
g (0) 3
x
6
( 6, 0)
(0,3)
e)
1
FALSE. cos
d) Increasing: ( 2, 2)
Decreasing: ( , 2) (2, )
e) Concave up:
Approximately ( 4, 0) (4, )
Concave down:
Approximately ( , 4) (0, 4)
5t 3
x
2h( x) 2
2
2x
9x 11
2
10x 8
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.
8.
3 5 11 13
,
,
,
8 8 8
8
a)
x
b)
x 1.18, 1.96, 4.32, 5.11
y
PART 2 – CALCULATOR
2 cos 2 x
y
A. Text
1.
a) Periodic
b) Not periodic
c) Not periodic
x
15