ion
tla
hat
CHAPTER 1
ion
PROBLEM SET 1.1
for
rst, econ
for
-st, ~c-
1. Acute, complement is 80°, supplement ...
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ion
tla
hat
CHAPTER 1
ion
PROBLEM SET 1.1
for
rst, econ
for
-st, ~c-
1. Acute, complement is 80°, supplement is 170° 3. Acute, complement is 45°, supplement is 135° 5. Obtuse, complement is -30°, supplement is 60° 7. We can't tell if x is acute or obtuse (or neither), complement is 90° x, supplement is 180 x 9. 60° 11. 45° 13. 50° (Look at it in terms of the big triangle ABC.) 15. Complementary 17. 65° 19. 180° 21. 120° 23. 60° 27. 15 25. 5 (This triangle is called a 3-4-5 right triangle. You will see it again.) 29. 5 31. 3V2 (Note that this must be a 45°--45°-90° triangle.) 33. 4 (This is a 30°-60°-90° triangle.) 35. I 37. v4I 39. 6 41. 22.5 ft 43. 2, v3 0
45.4,4V3 55. 8
6 _ 'h ,h v3 2v3,4v3
47.
57.
4 v2 =
63. a. xV2
2V2
b. xv3
.
49. 40ft
2
51.lOL6ft
61. a. V2 inches
59. 1,414 ft
53. b.
4V2 5
v3 inches
65. See Figure 1 on page 1.
PROBLEM SET 1.2 1. QIV 5.
3. QI
7.
y
... :
:It"
9. QII and QIll
11. QIlI
13.
y
;..
y
01
• x
X
(;-=.
y
13.
y
15. 8= 7Jr 3
8=_SJr 6
"' .'n
17. 2.11 1T
c
:J
.. x
19. 0.000291
21. 1.16 mi
21T
25. 6
27• - 3
3 33. ;
35.
41.
;;;;: 6((
~ lI. (,
29
3
37.
q
I
l>
x
l>
x
71T
23. 1 080 "" 0.02 radian ,
31. -21T 3
71T
• 4
1T 1T
I"
~
1T
+
1T
6
y
39.
21T -
!!... 4
y
43.
r
l>
60;)
x
~
I
I
:J
;=;:
60'~
Answers to Exercises and Chapter Tests
y
45.
y
47.
21.
29. 37. 45. 49. 59.
49. 57.3 0
51. 14.3 0
V;
67.
69.
~
%
71.
81. (0, -1), (f, 0), (rr, 1), 85. (0,0),
(~, 1), (f, 0).
89. (0,3), (f,2)' (rr, 1), 93.
o2
53.
73.
C;, C;,
4 77. 13
75. -2
(~, 2), (rr, 0),
83. (0, 0),
-1), (71",0)
87. (f, 0), (71",1),
(2rr, 3)
91.
I
61. -
C;,
C;,
-2), (271",0)
95.
rr
11
C;, C;,
0), (271", -1),
(-~,O), (0,3), (~,O), (f, -3).
(%,s) G,3)
63.0
(~, ~), (f, 1),
79. (0,0),
0), (2rr, -1)
C;,2),
(-~,3}(~'-2)'G,3)
V;
59. 2
57. -2
55.
65.
-2V2
C;, ~),
(rr, 0)
69. 85.
will b. ( leng
posi
0)
95.
101.
0) PR
97. 0.9965
1. 6 sinO
tan 0
cosO
cot 0
sec 0
csc 0
13.
3
99.
1
I
-
-3
VfO
3
I
101. 103.
VfO
2
2 2
1
y)
y)
1 2
y)
y)
-
37.
2
2
2
25.
3
49.
2
61.
PROBLEM SET 3.3
1 1 . 2
3. 0
PR
7 1
5. -2
• 2
11.
9.
-V2
1. 1 15..
13.
sinO
cosO
1
0
2
2 I
15.
-
17.
0
19.
2
1
V2
tan 0 1
0
0
sec 0
-0
2
-0
2 1
cot8
0
undefined
-1
-1
2
o -V2
csc 0 21.
2
para
-2
31.
undefined
39.. 47.
Answers to Exercises and Chapter Tests
21. !I". 57T 6' 6
23. 57T 77T 6 ' 6
29. sin 77T 6
77T
-0.5, cos ( ;
~
-0.8660
37. sin 75° ~ 0.9659, cos 75°
~
0.2588
45. esc e = 49. esc t
= 1.1884, sec t
=
31. 0.5236, 2.6180 39. 210°, 3300
V5 2: ' sec e V5, cot e =
1
47. sin t
2
1.8508, cot t
21
71. cosecant and cotangent
7T 4
"4 ~ 0.7071, cos
0.6421
73. No
1
2
~
0.8415, cos t
=
53.
35. sin 120° 43. sin
e=
-
~
0.8660, cos 120°
~, cos e
-0.5
1, tan e = -2
0.5403, tan t = 1.5575
e
X
55. A + B
57. 2
63. -1.6198
61. -0.7568
75. Yes
0.7071
33. 3.1416
41. 45°, 225°
51. 7T
... . 27T . 2;1 functIOn IS cosme, argument IS 3 ' value IS -
59. 69.
-
. 7T 27. 8m
25. 27T 57T 3 ' 3
79. No
77. No
7T
+ 8"
67. 1
65. 1.4353
81. Yes
83. No
85. The value of csc t is undefined at t = O. For values of t near O. csc t will be a large positive number. As t increases to 7T/2, csc t will decrease to 1. 87. The value of sin t will decrease from 1 to O. 89. See the Solutions Manual. 91. a. Close to 0 b. Close to 1 c. Close to 0 d. Becoming infinitely large e. Close to 1 f. Becoming infinitely large 93. The shortest length of DE occurs when point A is at either e = 0 or e = 7T, when this distance is the radius of the circle, or 1. For all other positions, DE is greater in length. 7T 7T 7T 7T 7T 7T 95. a. 3 b. (5 c. 3 d. 6 e'"3 f. (5 97. B 48°, a = 24,b 27 99. A = 68°, a 790,c 850
101. A
=
33.1°, B
56.9°, c
=
44.7°, B = 45.3°, b
103. A
37.5
4.41
PROBLEM SET 3.4 1. 6 in.
3. 2.25 ft
5. 27T em
47T 15. 9 ft ~ 1.40 ft
13. 4,400 mi
47T 7. 3 mm
6.28 em
~
9. 407T 3 in. ~ 41.9 in.
4.19mm
21. 2,100 mi
19. 1.92 radians, 1100
17. 33.0 feet
11. 5.03 em 23. 65.4 ft
29. 0.5 ft 33. 4cm 25. 480 ft 27. a. 103 ft b. 361 ft c. 490 ft 31. 3 in. 35.1 m 16 _ 2 . 2 97T Z _ 2 257T 2 2 37. 57T km - 1.02 km 39. 9 em 41. 19.2 m 43. 10 m - 2.83 m 45. 24 m ~ 3.27 m 49. 2 cm 61. 74.0°
51.• ~. in. v3 63. 62.3 ft
R::
2.31 in.
53. 9007T ftz
65. 0.009 radian
R::
2,830 ft 2
55. 3507T mm R:: 100 mm 57. 60.2° 59. 2.31 ft 11 67. The sun is also about 400 times farther away from the earth.
0.518°
R::
47. 4 in 2
PROBLEM SET 3.5 1. 1.5 ftlmin
15. 4 rad/min
3. 3 crnlsec 8
5. 15 mifhr
17. 3 rad/sec
7T 21. d = 100 tan 2t; when t
1
= 2' d =
~
47. 23.3 rpm
25. 1807T m
. 209 rad/mm
39. 807T ftlmin "" 251 ft/min 49. 5.65 ftlsec
100 ft; when t R::
3
= 2' d
11. 7 mi
9. 22.5 mi
19. 37.57T rad/hr
2.67 rad/sec
23. 40 ill.
parallel to the wall. 2007T. 31. -3- rad/mm
R::
7. 80 ft
=
R::
27. 4,500 ft
33. 11.67T rad/min "" 36.4 rad/min 41.
7T 12 rad/hr
R::
0.262 rad/hr
51. 0.47 mi/hr
53. h
27T 15 rad/sec
=
0.419 fad/sec
1, d is undefined because the light rays are 29. 207T rad/min
35. 10 ill.!sec
98.5 cos
G; t)
R::
62.8 rad/min
37. 0.5 fad/sec
43. 3007T ft/min "" 942 ftlmin 110.5
R::
118 rad/hr
-100 ft; when t
565 m
13.
45. 9.50 mifhr
Answers to Exercises and Chapter Tests
55. 889 rad/min (53,300 rad/hr) 65. 80.8 rpm
67.
Ivxl
57. 12 rps
54.3 ft/sec,
IVA
59. See the Solutions Manual. 40.9 ft/sec
61. 18.8 km/hr
63. 52.3 mm
5.
69. 71.9 mi west, 46.2 rni south
71. See the Solutions Manual.
CHAPTER 3 TEST y
1.
y
2.
9.
~----~+-~~------~x
3. -1.1918 9. -
15. 22. 27. 29.
1
v'3
4. 1.1964 10.
25'lT
18
6. 174.0°
5. -1.2991 11. 105°
12.
7. 226.0° 2
13.
8. 14. sin t
1
v'2 3
v'I3 ' cos t
2
4x 16. 0.2837 17. No 18. 0 19. 2'lTft R:: 6.28ft 20. 4 cm 21. 10.8 cm2 2 lO'lTft R:: 31.4 ft 23. 8 in. 24. 72 in. 25. 0.5 rad/sec 26. 80'lT ft/min 28. 2,700'lTftirnin R:: 8,480ftimin 4 rad/sector the 6-cm pulley and 3 rad/sec for the 8-cm pulley 22.0 rpm 30. 41.6 km/hr
, tan t
=
3 2
13. 23. 29.
CHAPTER 4 35.
PROBLEM SET 4.1 y
1. 2
--+-+-""k--+--+--+-I'--+-+~
y
3.
o
2x
2
x
~O+-1C+-+1C-3-t1C-t1C--If-t-+-~ x
-1
-1
-2
-2
- -- 424
For 49.
Answers to Exercises and Chapter Tests
5.
y
7.
L
2
2
• r
t
I
I
J~
\
x
I
J
\
!
I. x
1C
-I
-2
-2
11.
y
9.
y .7
• I -41C -31C -21C
21C
31C
• r.
I" x 41C
II:Iff"'\\1I: I
1T
31T
1T
23. 0,
17. 0, 1T, 21T
1
2
25.
1T, 21T
29. -.:.. 2
"2
15.
13. 2 ' 2
27..
'I
1T
19. 2
1T
31T
21. 2'-2
v'3 2
1 33. 3
31. y
35.
vr
-0/
)VI
iii
sin (180 0 -
•
(;/)
x
== y == sin (;/
For Problems 37 through 47, solutions are given in the Solutions Manual.
49. 60°
51. 45°
53. 120°
55. 330 0
,
rt' . i
~
r
,r
x
JL--:--++ x
Answers to Exercises and Chapter Tests
57.
3.5
59.
1.2
-2ff
-2ff
-1.2 The amplitude is decreased.
-3.5 The amplitude is increased. 61.
4
63.
65.
1.2
1.2
19.
o
-2ff
o
25. -1.2
-1.2
-4 The graph is reflected about the x-axis.
The period is doubled.
The period is halved.
PROBLEM SET 4.2 1.
y
y
3.
5.
y Amplitnde =
c', 1
t.
6 3
29.
-+--+--I--II---+----,l-l>- x
o
-3 -6
7.
y
9.
y
11.
y Period::::
7[
33.
Answers to Exercises and Chapter Tests
y
13.
15. Period
0'
t'tr
I~ x
-1
11: 4
0
-1 7T
24.
"6
28.
2
..,
%
25 _
.
hx
1
7T
4
7T
29. 6
26. 36.40
27.
~39.7"
30.
CHAPTER 5 1 For some of the problems in the beginning of this problem set we will give the complete proof. Remember, however, that there is often more than one way to prove an identity, You may have a correct proof even if it doesn't match the one you find here. As the problem set progresses, we will give hints on how to begin the proof instead of the complete proof, Solutions to problems not shown are given in the Solutions Manual. sin 11 cos 0
coso, -
1. cos 11 tan 11 =
sin 11
9. cos x(csc x + tan x)
cos x esc x =
cos x
+
.
SIll X
cos x
.,-- +
. smx
SIll X
=
cot X
+
sin x
cos x tan x
+ cos X
sin x cos
X
Answers to Exercises and Chapter Tests
3
cos2 t
sin2 t
sin 2 t
= cot2 t
- 1
19. Write the numerator on the right side as 1 - sin2 (J and then factor it. 25. Factor the left side and then write it in terms of sines and cosines. 27. Change the left side to sines and cosines and then add the resulting fractions. 33. See Example 6 in this section. 37. Rewrite the left side in terms of cosine and then simplify.
67.
71.
f
is one possible answer.
(J
-
e
TT 4 is one possible answer.
69. (J
F{
= 0 is one possible answer.
N
59
Note For Problems 73-79, when the equation is an identity, the proof is given in the Solutions ManuaL
65
73. Is an identity. 75. Not an identity; x = TT/3 is one possible counterexample. 77. Not an identity; A = '1T16 is one possible counterexample. 79. Is an identity. 81. See the Solutions ManuaL
87.
v3
91. 105°
2
4
5' tan A
83. cos A
3 4
85.
v3 2
93. See the Solutions ManuaL
PROBLEM SET 5.2 1.
yI6'-V2
3
4
9. sin (x
V6
V2
. --=::----=
+ 2'1T)
5.
=2
V6+V2
---~.
4
sin x cos 2TT + cos x sin 2TT + cos x(O) sin x
71.
= sin x(l)
For Problems 11-19, proceed as in Problem 9. Expand the left side and simplify. Solutions are given in the Solutions ManuaL
21. sin 5x 27.
23. eos 6x
y
25. cos 90°
29. sin 2x
0
y
31. y=
2x
y y=
(x
PF
1.
11.
..,.,
--------
.'~
Answers to Exercises and Chapter Tests
16 63 16 35. - 65 ' 65 ' - 63 ' QIV
y
33.
1
39. 1
37. 2,2,' QI
41. sin 2x
2sin x cos x
2t /1. l/
"- °1!'- 2;\ 3 -1
6
1
1 1
I J.. x
6"
7"
- 3
6
-2
For Problems 43-57, solutions are given in the Solutions Manual.
For Problems 59-63, when the equation is an identity, the proof is given in the Solutions Manual.
59. Is an identity.
61. Not an identity; x
y
65.
0 is one possible counterexample.
67.
l,
y
-1~ "4
,,\
: 1-· t'
Ii+- x i"
3"
2\ '4
.. 0
-1
I
,_
k
"
t
x
• oI ",\ ,
0/
I_ x
I '~2
-1
-3
-2
y
73.
y
75.
t
3
Tt
2
2 x
0
"6
-I
"3
" "2
- 180° 6. B = 71°, C = 58°, C = 7.1 ft; B' = 109°, C' = 20°, c' = 2.9 ft 7. 11 em 8. 95.7° 9. A = 43°,B = 18°,c = 8.1m 10. B = 34°,C = 111°, a = 3.8m 11.498em2 12. 52em2 14.51° 15. 410ft 16.14.1m 17.142mi 13. 17km2 18. 4.2 mi, S 75° W 19. 90 ft 20. 300 mifhr or 388 miJhr 21. 65 ft 22. 260 miJhr at 88.9° from due north 27. 98.6°
26. -8
23. 13
28.VoW=0
24. -5i . 5
29. b
+ 41j
3
= -
25.
vm
30. 1,808
CHAPTER 8 PROBLEM SET 8.1
1. 4i
2 15 . x = -5' Y = -4 7T
7T
37. -1
39. 1
49. 41
51. 53
5 12 59. - - + - i 13 13
57T
23. 10 - 2i
+ 2i
31. 2
33. 12
43. -48 - 18i
53. -28
+ 4i
55. 4 63. 61
69. 10 - 3i
71. 16
75. See the Solutions Manual.
.
•
+ 6i
+ 2i
+ 20i
+
2rV3; x
b
~,eos()
93. B = 69.6°, C = 37.3°, a
= 1 -
85. Yes
83. See the Solutions Manual.
va2 +b2
57T
or -
4'
Y
1
= -
2
7T
=
2
35. 1 47. 5
+ 12i
+9
77. See the Solutions Manual.
81. x = 1
. 89. sm() =,
4
2
3' Y
27. 3 - 13i
73. x 2
+ 2i, Y = 4 - 2i; x = 4 - 2i, Y = 4 + 2i 2 -
7T
= -
13. x =
45. 10 - 10i 1 3. 57. - + -1 -6 5 5
17
+ - i 65. 13 61
79. x = 4
rV3, Y =
11. -3
19 x
25. 2
41. i
61. -2 - 5i
+ 22i
9. -6
17. x = -2 or 3, y = ±3
21 x = - Y = - or . 2' 4 4 29. 5 eos x - 3i sin Y
67. -7
7.2rV2
5.3rV2
3. 11i
=,
a
~
va 2 +b2
=
248 em
rV3, Y =
2
+ 2rV3
. 4 3 87. sm () = - - eos () = - 5' 5 91. 135° 95. A = 40.5°, B = 61.3°, C = 78.2°
1
2
3
3
Answers to Exercises and Chapter Tests
y
1.
3.
• x
~.
_ x
y
19.
• x
V3 + i
27. 9.78
~
21. -2
2.081
-t-
2iV3
29. -80.11
23.
2
+ 59.851
35. Vi(cos 13SC + i sin 135°) = Vi cis 135° 37. V2(cos31SC
+ isin315°)
cis 315°
1
-/
25.
2
Vi cis 3: Vi cis 7:
X
vi2 vi2 i
31. -0.91 - 0.42i =
to
y
"~K
17.
',"~"r+~~;
V3
.. x
.. x
y
~
y
x
11.
,~
15.
~.
•
y
9.
......-,--'-----+-li,~"~, .
13.
r
~
y
7.
5.
33. 9.60 - 2.79/
'.
x
J
X
y
')!II~'.
Answers to Exercises and Chapter Tests
41. 8(cos 90°
+ i sin 90°)
•
8 cis 90°
1
1T
8 CIS 2
43. 9(cos 180°
+ i sin 180°)
9 cis 180° = 9 cis 1T
45. 4(cos 120°
+ i sin 120°)
4 cis 120°
4 cis 2;
+ i sin 53.13°) 5 cis 53.13° 49. 29(cos 133.60° + i sin 133.60°) 29 cis 133.60° 51. 25(cos 286.26° + i sin 286.26°) 25 cis 286.26° 53. 17(cos241.93° + isin241.93°) = 17 cis 241.93° 47. 5(cos 53.13°
For Problems 55-61, see the answer to the corresponding problem. For Problems 63 and 65, see the Solutions Manual. 7 6
V6·
56 69. 65
4
77. B
62.7°
B
or
=
7
\13
71. sin 120°
73. cos 50°
2 79. Answers will vary.
117.3°
75. No triangle is possible. 1
1:
l'
PROBLEM SET 8.3
+ i sin 50°) 3. 56(cos 157° + i sin 157°) 5. 4(cos 1T + i sin 1T) 2(cos 180 + i sin 180°) 9. -2\13 2i = 4(cos 210° + i sin 210°) 12(cos 360° + i sin 360°) 13. -4 + 4£ = 4V2(cos 135° + i sin 135°)
1. 12(cos 50° 7. -2
11. 12
15. -5 - 5i\13
1O(cos 240°
=
+ 32i\13
19. 32
27. -4
29. -8 - 8i\13
41. 2(cos 0°
+ i sin 0°)
45. 2[ cos ( - 270°)
49. -4
+ i sin 240°)
21. - 81 - 81 2
23. -
2
+ i sin 19°)
43. cos (-60°)
+ i sin ( -270°)]
2i
2'
25. 32i
39. 0.5( cos
+ i sin ( -60°)
47. 2[cos (-180°)
f + i f)
+ i)-I
61 _ 7
9
67 4V2 • 7
63.
[V2(cos45°
+ isin45°)r l = ~ - 1 i
V6
V6
3
69. 6.0mi
65.
2
'2 71. 103° at 160 rniJhr
2
59.
sin
=
0.5 cis
f.
\13.
--I
2
2
+ i sin (-180°)]
51. 8
4i
2:
33. 16 + 16i
37. 1.5(cos 19°
2
~ + ~i
For Problems 53 and 55, see the Solutions Manual.
·
4 cis 1T
17. See the Solutions Manual.
~i
31. -8i
35. 4(cos 35° + i sin 35°)
57. (1
=
0
\13 + -1.I 4
4
=
-2
31
Answers to Exercises and Chapter Tests
1.
y
3.
",~"'f·T
V3 + i,-V3
7.
11 5" -5' •
1,
13
1
'. x
vi + lVi, - vi
9.
•
~
\21+62 ,t, V6 2
\
• x
lVi
Vi, 2 I
15. 2(C0870° + i sin 70°), 2(C08 1900 + i sin 190°), 2(C08 310° + i sin 310°) 17. 2(cos 10° + i sin 10°), 2( cos 1300 + i sin 130°), 2( cos 250 0 + i sin 250°)
19. 3(cos 60° + i sin 60°), 3(cos 180° + i sin 180°), 3(C08 300 0 + i sin 300°)
21. 4( cos 30° + i sin 30°), 4(cos 150 + i sin 150°), 4( cos 270 + i sin 270°) 0
23. 3, _ 3 + 2
3V3 , 2 l,
3 2
3V3 2
0
25. 2, -2, 2i, -21
i
27.
vi + i, -1 + lvi, V3 - i, 1 - iV3
29.
+ i sin 3°) 1O(cos 75° + i sin 75°) 10(cos 147 + i sin 147°) 1O(cos219° + i sin 219°) 1O(cos 291 + i sin 29JO) lO(cos 3° 0
0
31.
+ 0.52i ;::;; 2.59 + 9.661
;::;; -8.39 + 5.451
;::;; 9.99
;::;; -7.77 - 6.29i
;::;; 3.58
9.341
y
i
+ i sin 0) where 0 (cos 0 + i sin 0) where 0
33. Vi(cos 0 35.
vi
y
5.
30°,150°,210°,330° = 67.5°,112.5",247.5°,292.5°
i'
n, :
•
X
Answers to Exercises and Chapter Tests
y
37.
y
39.
)' = -2 sin (-3x)
2 ,in 3x
because sine is an odd function.
41.
Y
3 2
~o~--+-~--~--+-~~x
-1
-2
-3
43. 4.23 cm2
45. 3.8 ft2
47. -3.732, -0.268,4.000
PROBLEM SET 8.5 y
1.-11. (odd)
7
13. (2, -300°), (-2,240°), (-2, -120°) 17. (-3, -330°), (3, -150°), (3,210°) 23. (-1, -1)
25. (-6, -2\13)
33. (4,150°)
35. (2,0)
15. (5, 19. (1,
\13)
(-5,315°), (-5, -45°) 21. (0, -3)
27. (1.891,0.65lJ)
37. (2, 7:)
39. (5,53.1°)
29. (-1.172,2.762) 41. CVs,l16.6°)
31. (3v2,135°)
Answers to Exercises and Chapter Tests
43.
(V13, 236.3°)
49. x 2
+
l
45. (9.434,57.99°) 51. x 2
= 9
5 57. r =
l
53. (x 2
6y
+ l)2
= 8x y
55. x
y
3
_ .
cosO - sme
~9. r
2
61. r
y
65.
47. (6.083, -99.46°)
67.
6 cosO
=
63. 0 = 45°
or
y
cos 0
sin 0 y
69.
6
6
1
5
3
4
x
3
x
2
'j
-3
-6
I_ x 11:
11:
311:
211:
2
2
PROBLEM
1.
y
3.
5.
y
y '->("";;
;. x
7.
y
9.
x
y
I~.
•
11.
".,. x
• x
y
.,. x
Answers to Exercises and Chapter Tests
15.
y
13.
y
17.
y
3
,6.9\1",
y
19.
21.
y
23.
y
8
4~
27.
y
25.
29.
x
53 31.
33.
35.
8,-,
9,-,
T
O
x ~~~b,-,-;-,~--~,-_-,