212
PART I: Solutions to Odd-Numbered Exercises and Practice Tests
Solutions to Odd-Number~ Exercises
x
lo y = sec2
Period:
3. y = tan 2x
2qr
1¢2
- 4~r
Period: -2
Matches graph (f).
Matches graph (g).
7. y = -csc
cot2
Period:
Period:
-2
Matches graph (e).
Matches graph (b).
9. y = 1~tanx
11. y = .2tan2x
i~eriod: ~r
Period:
Two consecutive asymptotes:
Two consecutive asymptotes:
x=-
and x = 2
2x
0
x
y
1
0
.-~ x -2
4
2x = -~ ~x 4
1
y
X
0
Y 2
0
-2
y
1
13. y=-~secx
1
Graph y = - ~ cos x first.
Period:
One cycle: 0 to 2"tr
15. y = -sec ~rx
Graph y = -cos ¢rx first.
Period: 2_ff_~ = 2
One cycle: 0 to 2
III
n-2 n-1 n
I
213
PART 1: Solutions to Odd-Numbered Exercises and Practice Tests
x
17. y = sec ,rrx - 3
19. y = csc2
Reflect the graph in Exercise #15 about the
x-axis and then shift it vertically down three units.
x
Graph y = sin ~ first.
27r
Period: 1-~ = 4~r
One cycle: 0 to 4,rr
1_3,
I
-4
I
!
!
!
1
!
!
X
21. y = ~ cot
Y
2
Period: ,tr = 2-tr
1/2
Two consecutive asymptotes:
x
-=0 ==>x=0
2
x
- = "rr ==> x = 2,rr
2
Y
1
0
1
1
23. y = ~ sec 2x
1
Graph y = ~ cos 2x first.
2~r
Period: ~ =
One cycle: 0 to
214
PART I: Solutions to Odd-Numbered Exercises and Practice Tests
25. y=2tan-4
Period: - 4
,rr/4
Two consecutive asymptotes:
x
-1
0
1
Y
-2
0
2
27. y = csc(~r- x)
"U
Graph y = sin(’n" - x) first.
I
I
Period: 2~r
I
I
I
Shift: Set "rr- x = 0 and "tr- x = 2~r
-2n
X-- --’Tr
29. y = 2 cot
Period: ~r
Two consecutive asymptotes: x - _-- = 0 ==> x =
2
,n"
2
2
2
3,rr
I
!
I
I
!
I
I
!
I
I
!
I
I
!
I
I
I
I
I
I
!
I
X -- ~ -" q’~ :=:::~X -- --
3qT
\ ,-n\ ,
i I\’ I n\, I\
\:
-4-
5’~
x
2
Ix
31o y = -~tan-~
33. y = -2 sec 4x
-2
cos 4x
-1.57
-4
I
1.57
I
\’, \
I
215
PART I: Solutions to Odd-Numbered Exercises and Practice Tests
1
39. y = ~ sec(2x- "n’)
37° y=~cot x+
1
.__
1
Y = 2cos(2x - "rr)
4tan(x + ~)
-4.71
41. tan x = 1
43. secx= -2
7"n" 3"rr ~r 5-rr
4’ 4’ 4’ 4
y
2"rr 4"rr
x=+---~-’ + 3
y
-. 1: /1’, ,’,
I~ I~ 1/;,,
/’,1 !/I i/~ u/~
~J,,
,’ ! ~’-~,’"--,
,, ILL -~’~
I
I
¢r ’~ 2~r
!
!
45. The graph off(x) = sec x has y-axis symmetry. Thus, the function is even.
47. Yl=sinxcscx and Y2= 1
-8
i
i
t
i
cos x
1
49. y~
- sin xand y2 = cot xtan
- x
3
-2
Not equivalent because yl, is not defined at 0
Equivalent
sin xcscx = sin X(sin-~)= 1, sin x :/= 0
cot x -
Sl. f(x) = xcosx
COS X
sin x
a(~) = I~1 sin x
As x ~ O, f(x) ~ O.
As x ~ O, g(x) ~ O.
Matches graph (d).
Matches graph (b).
216
PART I: Solutions to Odd-Numbered Exercises and Practice Tests
1
57. f(x) = sinz x, g(x) = ~ (1 - cos 2x)
f(x) = g(x)
The graph is the line y = 0.
2
2
/k/\
59. f(x) = e-x cos x
Damping factor: e-x
6.28
61. f(x) = "2-x/4 cos "n’x
-2-x/4 < f(x) < 2-2x/4
3
The damping factor is y = 2-x/4.
6
-3
As x ~ ~, f(x) --~ O.
-6
As x ----> c~z, f---> 0
6
63. y=-+cosx
65.
2
As x ---> O, from the right, y --> o~.
As x --~ 0. from the left, y--~
6
sin x
As x tends to 0, ~ approaches 1.
x
"-25.13
5
69. tan x = d
67. f(x) - tan x
Asx -~ O,f(x) ----~1
d36
-3.14 ~ ~--~3.14
-6
5
- 5 cot x
tan x
217
PART I: Solutions to Odd-Numbered Exercises and Practice Tests
71. As the predator population increases, the number of
prey decrease. When the number of prey is small, the
number of predators decreases.
73° (a) 0.~
(b) The displacement function is approximately
periodic, but damped. It approaches 0 as t
increases.
~t
75. S=52+5t-28cos-6
(a)
s
77. (a) If a spring of less stiffness is used, then c will
be less than 8.2.
(b) If the effect of friction is decreased, then b
will be greater than 0.22:0.22 < b < 1.
oa
2 4 6 8 10 12
Month (1 ~-> January)
(b) least sales: January
(t = 1, S ~ 32.75 thousand units)
greatest sales: June
(t ~ 6.66, S ~ 111.64 thousand units)
3qr
~" x = -~- isqr
79. True. -~- + qr = -~- and
a vertical asymptote for the tangent function.
81. True. 2x sin x -~ 0 as x ---> - ~.
83. Forf(x) = csc x, as x approaches ,rr from the left, f approaches c~. As x approaches ~"from the right, f
approaches - ~.
85. (a) y, =
Y2 =
(b) Y3 =
(C)
sin(qrx) + ~sin (3~rx)
sin(~rx) + ~ sin (3,rrx) + ~ sin (5qrx)
sin(,trx) + ~sin (3~rx) + ~ sin (5~rx) + ~ sin (7~’x)
Y4-" sin(qrx) + ~sin (3~rx) + ~sin (5qrx) + ~sin (7~rx) + ~sin(9~rx)
218
PART I: Solutions to Odd-Numbered Exercises and Practice Tests
89. One-to-one.
87. Not one-to-one
y= 3~-5
x3=y- 5
y=x3+5
f-l(x)= x3 + 5
91. hyp=-,/~-+ 142= ~
sin0- ~
9
14
cos0- ~
9
14
tan 0 = -cot 0 = -14
9
csc0- ~
sec0- ~
9
14
Section 4.7
Inverse Trigonometric Functions
[] You should know the definitions, domains, and ranges of y = arcsin x, y = arccos x, and y = arctan x.
Function
Domain
Range
y=arcsinx ~ x=siny
-1< x<l
y=arccosx ~ x=cosy
-1< x<l
---<y<-2- -2
0 <y< "rr
y=arctanx ~ x=tany
---<y<-22
[] You should know the inverse properties of the inverse trigonometric functions.
sin(arcsin x) = x, - 1 _< x <_ 1 and arcsin(sin y) = y, --~ < y < --2
cos(arccos x) = x, - 1 < x < 1 and arccos(cos y) = y, 0 < y < "n"
tan(arctan x) = x and arctan(tan y) = y, --~ < y < 2
You should be able to use the triangle technique to convert trigonometric functions or inverse trigonometric
functions into algebraic expressions.