219
PART I: Solutions to Odd-Numbered Exercises and Practice Tests
Solutions to Odd-Numbered Exerdses
1. (a)
-1.0
x
-0.8
-0.6
-0.4
-0.2
Y -1.5708 -0.9273 -0.6435 -0.4115 -0.2014
x
0
0.2
0.4
0.6
0.8
1.0
y
0
0.2014
0.4115
0.6435
0.9273
1.5708
(c)
(b)
3. (a)
-10
x
-8
(d) (0, 0), Symmetric to the origin
-6
-4
-2
Y -1.4711 - 1.4464 -1.4056 - 1.3258 -1.1071
x
0
Y 0
(b)
2
4
6
8
10
1.1071
1.3258
1.4056
1.4464
1.4711
(c)
y
(d) Horizontal asymptotes: y = +-2
2
.-10
-10 -5
5 lO
-2
5. tan ~)=-l~arctan(-1)- ~r4
7. arcsin(- 1) = -7 ~ sin -
9. (a)y = arccos~cosy = ~for0<y<’n’~y = 3 ~ 1.047
(b) y=arccos0~cosy=0for0<y<-rr~y=--~ 1.571
2
11. (a)y=arccos(--~-)~cosy2-(b) y = arcsin
13. (a) y = arccos
- -~for0 < y < "n’~y=m~2.356
4
sin y - - ~ for ~
-~ --~< y<-~,y=-~-~
~r
"n" - 0.785
~ cos y = -~ for 0 < y < ~r ==> y = --f- ~ 2.094
(b) y = arcsin-~-- ==> siny = -~- for--~<y<-~==>y = 4 0.785
=-1
220
PART I: Solutions to Odd-Numbered Exercises and Practice Tests
¯
"rr
~
¢r - 1.571
15. (a) y=arcsin(-1)~siny=-lfor-7 < y < ~ y=-7~
(b) y=arccosl~cosy= lforO < y< ~y=O
17. y = arccos x
x = cos y
19. (a) arcsin(- 0.75) ~ -0.85
(b) arccos(-0.7) ~ 2.35
21. (a) arcsinO.41 ~ 0.42
23. (a) arctan 0.98 ~ 0.78
25. f(x) = sin x
g(x) = arcsin x
(b) arctan 4.7 -~ 1.36
(b) arccos 0.36 ~ 1.20
y=x
4
27. cos 0 = -
x+l
10
Ix+ 1\
0 = arctan/---i--O--)
29. tan 0 -
x
4
0 = arccosx
31. tan (arctan 35) = 35
33. sin (arcsin (-0.1)) = -0.1
35. arccos cos- = arccos 0 = ~-
37. arcsin (sin ]ff) = arcsin
3
39. Let u = arcsin 5’
qr
3
sinu = "~, 0 < u <~.
=secu- adj -4
41. Let u = arctan - ,
tan u =
12
5’
csc arctan -
~
< u < O.
2
= csc u -opp - 12
221
PART I: Solutions to Odd-Numbered Exercises and Practice Tests
.6
45. Let u = arctan ~,
11
sinu ..... <u<O.
8’ 2
tan arcsin -
6
qr
tanu = -i-i-, 0 < u < ~.
= tan u = ~ -
47. Let u = arctan x,
55
cot arctan
= cot u -
opp 6
49. Let u = arcsin(x - 1),
x--1
sin u = x -- 1 -1
x
tan u = x = -.
1
1
sec[arcsin (x - 1)] = sec u -
sin(arctan x) = sin u = op_2
hyp
x
1
51. Let u = arctan-,
hyp 1
adj - ,/2X- x~
x-h
53. Let u = arcsin ,
1
tan u = -.
sin u -
x
x-h
x-h
cot arctan = cot u -
opp
--x
-" COS/~ =
~/r2 - (x- h)2
r
222
PART I: Solutions to Odd-Numbered Exercises and Practice Tests
~/36 - x2
57. If arcsin
- u,
6
~36 - xz
then sin u =
6
x
Asymptote: x = 0
These are equal because:
x
arcsin
~/36 - x2
x
- arccos6
6
x
Let u = arccos 2
tan(arccos ~) = tan u -
59. If arccos
x-2
2
then cos u -
X
61. y = arcsin2
Domain: -2 _< x _< 2
- u,
x-2
2
Range: -~ _< y < --2
1.57
/
x-2
arccos
2
x
2
- arctan
~/4x - x2
x-2
63. g(t) = arceos(t + 2)
3.14
Domain: - 3 < t < - 1
This is the graph of y = arccos t shifted two units to the left.
o
223
PART I: Solutions to Odd-Numbered Exercises and Practice Tests
f(x) = ¢r + arctan x
Domain: (-
x
67. f(x) = arccos4
Domain: [-4, 4]
Range: [0, ,n’]
,3.14
6.28
o
69. f(t) = 4 cos ~t + 3 sin "rrt
= ./42 + 32 sin ~t + arctan
= 5 sin(~rt + arctan ;)
6
-6
ThegraphsuggeststhatAcosoot+Bsinoot=./A2+b2sin(wt+arctan~)istme.
71. (a) tan 0-
s
750
(b) When s = 400, 0 = arctan \7--f6/~-" 0.4900 (~ 28.07°)
Whens = 1600, 0 = arctan (17@00) ~ 1.1325 ( ~ 64.89°)
73. (a) O= arctant2~)~ 26.0o/n\ (0.45rad)
h
(b) tan (26.0°) = 5--d
h = 50 tan (26.0°) ~ 24.4 feet
41
41
224
PART I: Solutions to Odd-Numbered Exercises and Practice Tests
75. Area = arctan b - arctan a
(a) a=0, b= 1
Area = arctan 1 - arctan 0 =_~r _ 0 = ~
4 4
(c) a=0, b=3
Area = arctan 3 - arctan 0
~ 1.25 - 0 = 1.25
(b) a=-l,b= 1
Area = arctan 1 - arctan(-1)
(d) a=-l,b=3
Area = arctan 3 - arctan(-1)
= 1.25
77. (a) tan0-
x
20
x
0 = arctan20
(b) When x = 5,
5
0 = arctan ~ ~ 14.0°, (0.24 rad).
12
When x = 12, 0 = arctan ~ ~ 31.0°. = (0.54 rad.).
sin x
79. False. tan x = ~ ¯
COS X
81. y = arcsec x if and only if sec y = x where x < - 1 t_j x >_ 1 and 0 < y < "n’/2 and ,n’/2 < y <
The domain of y, arcsec x is (-c~, -1] U [1, ~) and the range is [0, "rr/2) U (’rr/2, ~r].
-2
-1
t ...........
83. (a) y = arcsec ~ ==> sec y = ,f~ and 0 <_ y < _-- U _-- < y _< ,rr ==> y = -:
2 2
4
(b) y = arcsec 1 ==> secy = 1 andO<y<~t_J-~<y<,rr=:~y = 0
5qr
(c) y = arccot(-.4r~) ==> ~ot y = - V~ and 0 < y < "rr ==~ y - 6
(d) y = arccsc 2 ==,, csc y = 2 and -~ < y < 0 U 0 < y < -~ ==> y 6
225
PART I: Solutions to Odd-Numbered Exercises and Practice Tests
y = arctan(-x)
85.
tan y = -x, --2 < Y <
z
87. Y2=~--Yl
1
arctan x + arctan- = yl + Y2
2
x
-tan y = x
tan(-y) = x, --2 < -y <
2
arctan(tan(-y)) = arctan x
-y = arctan x
y = -arctan x
Yl
X
X
89. arcsin x = arcsin ]- = arctan ~ _ x2
91. -585° coterminal with 720° - 585° = 135°.
Quadrant II
Reference angle 0’= 45°
sin (- 585°) ~
93.
19~r
19~r 5~r
coterm!nal with 6~r - 4 - 4 "
Quadrant III
Reference angle 0’ = -4
cos (- 585°) = 2
tan (-585°) = - 1
cos
-~ =
2
4
95. y=2-cos(x+
y
Asymptotes: x = 0, x = 2¢r, x = -2~r,...
6o
2.
-2-