Trigonometry Section 3 – Solve Application Problems using Right Triangle Trigonometry.
MATH 152 – COLLEGE ALGEBRA AND TRIGONOMETRY
Transcript of 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) 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)
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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