263
PART I: Solutions to Odd-Numbered Exercises and Practice Tests
83. False. There might not be . 85. 124°= 124°(~-~oo)
periodicity, as in the equation
~ 2.164"’"v ,radians
sin(x2) = 0
14
14
89. tan 30°= -- ~ x- tan 30°
x
14
= 24.249
~
3
87. -0.41°= -0.41(~-~oo)
.~ - 0.007 radians
16
16
~ 24.892
91. sin 40°= -- ==> x- .
x
sin 40°
95. f(x) = ~cot y
x
Section 5.4 Sum and Difference Formulas
[] You should memorize the sum and difference formulas.
sin(u + v) = sin u cos v + cos u sin v
cos(u + v) = cos u cos v ¥ sin u sin v
tan u + tan v
tan(u + v) = 1~ tan u tan v
[] You should be able to use these formulas to find the values of the trigonometric functions of angles whose
sums or differences are special angles.
[] You should be able to use these formulas to solve trigonometric equations.
Solutions to Odd-Numbered Exercises
1. (a) cos + = cos -~ cos ~ - sin ~2222
(b) cosZ + cos-y = -~- 2
2
264
PART I: Solutions to Odd-Numbered Exercises and Practice Tests
’rr_l (b)~sin7’n" ~__ 1 ~ -1 - ~
~-~ = sin 6 2
-7-- sin3 -~ 2 -
~) =
2
5. (a) cos(O° + 135°) = cos 0° cos 135° - sin 0° sin 135°
= cos 135° (b) cos 0° + cos 135° = 1
2
2
7. (a) sin(315° - 60°) = sin 315° cos 60° - cos 315° sin 60°
=
(b) sin 315° _ sin 60° -
22
2
2
2
2
-
9o sin 75° = sin(30° + 45°)
= sin 30° cos 45° + sin 45° cos 30°
22
2
4
2
11. sin 105° = sin(60° + 45°)
= sin 60° cos 45° + sin 45° cos 60°
2
2
-~(~ +
4"
2
2
2
- "!~’(w’~ + 1)
cos 75° = cos(30° + 45°)
= cos 30° cos 45° - sin 30° sin 45°
cos 105° = cos(60° + 45°)
= cos 60° cos 45° - sin 60° sin 45°
tan 75°= tan(30° + 45°)
tan 30° + tan 45°
1 - tan 30° tan 45°
tan 105° = tan(60° + 45°)
tan 60° ÷ tan 45°
1 - tan 60° tan 45°
_ !-,/~/3)+ 1 .f~ + 3
1 -(~/3/3) 3 6.,/~ + 12 v’~ + 2
6
3+4
1 - .,/~ - 1 - -,/~
_ 4+2.¢/~__2_ ~
-2
265
PART I: Solutions to Odd-Numbered Exercises and Practice Tests
13. sin 195° = sin(225° - 30°)
= sin 225° cos 30° - sin 30° cos 225°
= -sin 45° cos 30° + sin 30° cos 45°
cos 195° = cos(225° - 30°)
= cos 225° cos 30° + sin 225° sin 30°
= -cos 45° cos 30° - sin 45° sin 30°
tan 195° = tan(225° - 30°)
tan 225° - tan 30°
1 + tan 225° tan 30°
tan 45° - tan 30°
1 + tan 45° tan 30°
llqr
.15. sin-- = sin +~ +
12
= sin ~- cos ~ + sin -~ cos
_
COS-i-~- = COS
+
3"n" ~r
3¢r ¢r
= cos -~- cos ~ - sin ~ s~ ~
llg
tan -- = tan
4
+
tan(3cr/4) + tanCcr/6)
1 - tan(3~r/4) tan(qr/6)
-1 + (~/~/3)
1 - (-1)(./~/3)
-3+./5
3+45
__ -12 + 6v.~_/-ff
6
17.
12 6 4
sin -
= sin -
= sin -~ cos + - sin -~ cos -~
.45_
22
cos -
= cos -
2
2
4"
= cos ~- cos -~ + sin + sin 4
2
2
22
tan = tan (~2) (~ ~) tan(,rr/6)= 1 + tan(o/6)
tan(,rr/+
tan(’a-/4)
19. cos 40° cos 15° - sin 40° sin 15° = cos(40° + 15°) = cos 55°
21. sin 340° cos 50° - cos 340° sin 50° = sin(340° - 50°) = sin 290°
- -2 + 45
PART I: Solutions to Odd-Numbered Exercises and Practice Tests
tan(325° - 86°) = tan 239"
25. sin 3 cos 1.2 - cos 3 sin 1.2 = sin(3 - 1.2) = sin 1.8
cos -~ cos ~ - sin ~ sin ~ = cos +
12,n"
35
= COS ~
29.
Yl
0.4
0.6
0.8
1.0 "
1.2
1.4
0.9801 0.9211 0.8253 0.6967 0.5403 0.3624 0.1700
Y2
0.9801 0.9211 0.8253 0.6967 0.5403 0.3624 0.1700
x
0.2 ,
=sin cosx+sinxo cos2
COS x
= Y2
31.
x
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Yl
0.6621 0.7978 0.9017 0.9696 0.9989 0.9883 0.9384
Y2
0.6621 0.7978 0.9017 0.9696 0.9989 0.9883 0.9384
= sin ~ cos x + sin x o cos
6
2
= - cos x + -- sin x
2 2
= ½(cos x + ~’~ sin x)
0~1,5
= Y2
0
33.
x
Yl
Y2
2
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0.9605 0.8484 0.6812 0.4854 0.2919 0.1313 0.0289
0.9605 0.8484 0.6812 0.4854 0.2919 0.1313 0.0289
...............
y, = cos(x + ~r)cos(x = (cos x " cos qr- sin x ¯ sin
[cos x cos ~r + sin x sin ~r]
= I-cos x] I-cos x]
= COS2 X
= Y2
0~1.5
267
PART I: Solutions to Odd-Numbered Exercises and Practice Tests
For Exercises 35-37, we have:
sin u = ~, u is in Quadrant I ==> cos u - i~
cos v = -53-, v is in Quadrant II :=~ sin v = ~
37. costu" + v)" = cos u cos v - sin u sin v
35. sin(u + v) = sin u cos v + cos u sin v
+
-
12 4
_33
12 3
_ -56
-- 65
-- 65
For Exercises 39-41, we have:
sin u = ~s, u ~s in Quadrant II ==> cos u = =~
4
3
cos v = g,
v ~s in Quadrant
IV ==> sin v = -3
39. cos(u + v) = cos u ° cos v - sin u ° sin v
-96 + 21
125
43. cos(,rr - 0) + sin "n" +
-75
125
41o sin(v - u) = sin v ¯ cos u - sin u ¯ cos v
-3
72 - 28
125
5
44
125
= cos "rr cos 0 + sin ~ sin 0 + sin -~ cos 0 + sin 0 cos ~2
= (-1)(cos O) + (O)(sin O) + (1)(cos O) + (sin 0)(0) = -cos 0 + cos 0 = 0
tan x + tan ~r
tan ’n" - tan x
45° tan(x + ~r) - tan(qr - x) = 1 - tan xo tan ~r-
1 + tan ¢r tan x
= 2tanx
47° sin(x + y) + sin(x - y) = sin x cosy + sin y cos x + sin x cos y - sin y cos x = 2 sin x cos y
49. cos(x + y)cos(x - y) = [cos x cos y - sin x sin y][cos x cos y + sin x sin y]
= cos~ x cos2 y - sin2 x sin~ y
= cos~ x(1 - sin~ y) - sin: x sin2 y
= cos2 x - sin2 y(cos:~ x + sin2 x)
= cos2 x - sin2 y
268
PART I: Solutions to Odd-Numbered Exercises and Practice Tests
51. sin(arcsin x + arccos x) = sin(arcsin x) cos(arccos x) + sin(arccos x) cos(arcsin x)
=~.x+ ~/~- x~ ¯ ~/~-x~
= x2 + 1 x2
=1
0 = arcsin x
0 = arecos x
53. Let:
u = arctan 2x and v = arccos x
tan u = 2x
cos v = x
1
x
sin(arctan 2x- arccos x) = sin(u - v)
= sin u cos v - cos u sin v
2x
1
,/4x~ + ~(x)
~:~x~ - ./i - x~
,/4x~ + 1
55.
sin(x + ~) + sin(x - ~) = 1
sin x cos ~ + cos x sin ~ + sin x cos ~ - cos x sin ~ = 1
2 sin x(0.5) = 1
sin x = 1
2
269
PART I: Solutions to Odd-Numbered Exercises and Practice Tests
57.
cos x cos ~- - sin x sin ~ - cos x cos ~- + sin x sin = 1
-2sinx -~- = I
-~,/~sinx = 1
sin x =
sin x =
X--
1
~:
./2
2
5¢r
4’ 4
tan(x + zr) + 2 sin(x + or) = 0
59.
tan x + tan "n"
+ 2(sin x cos ,r + cos x sin or) = 0
1 - tan x tan ~r
tan x + 0 + 2[sinx(-1) + cosx(0)] = 0
1 - tan x(0)
tan x
- 2 sin x = 0
1
sin x
- 2sinx
COS X
sin x = 2 sin x cos x
sin x(1 - 2 cos x) = 0
sin x = 0
or
x=O,~
0
1
2
7r 5qr
x-3,3
COS X -- --
6.28
The poims of intersection occur at x = 0.7854 and x = 5.4978.
270
PART I: Solutions to Odd-Numbered Exercises and Practice Tests
63. tan(x + 7r) - cos(x + ~) = 0
I
I
I
The points of intersection occur at x = 0 and x ~ 3.14.
65. Yl +yz=Acos27r t _
+Acos2cr +
= A[cos(~)cos(~-~)+ sin(~)" /2¢rx’~] A[cos(~) cos(~)
= ~ cos
cos 2
69. False. sin 75° = sini30° + sin 45°)
= sin 30° cos 45° + cos 30° 45°
67. False. See page 404.
22
22
+
--
4
71. cos(nqr + 0) = cos n’rr cos 0 - sin n~" sin 0
= (- 1)n (cos O) - (O)(sin O)
= (- 1)’~ (cos 0), where n is an integer.
b
b
73. C=arctan- ~ tanC- --o sinCa
a
b
a
bz’ cost- ~/a2 + b2
J~2 +
a
-,/az+bzsin(B0+ C) = ~/a2+ b2 sinB0 ° ~/a~+bZ + ,/,a2+bZ ° cosB0 = asinB0+bcosB0
(
)
75. sin 0 + cos 0
a=l,b=l,B=l
b
,n
(a) C = arctan-= arctan 1 = -a
4
sin 0 + cos 0 = ./a~ + ba sin(B0 + C)
a
,/r
(b) C = arctan~- = arctan I b
4
sin 0 + cos 0 = x/a2 ÷ bz cos(B0 - C)
77. 12sin30+5cos30
a= 12, b=5, B =3
b
5
(a) C = arctan- = arctan s-2 -~ 0.3948
a
IZ
12 sin 30 + 5 cos 30 = ~/a2 + b2 sin(B0 + C)
~ 13 sin(30 + 0.3948)
a 12
(b) C = arctan ~- = arctan -z- ~ 1.1760
12 sin 30 + 5 cos 30 = ~/a2 + b2 cos(B0 - C)
~ 13 cos(30 - 1.1760)
271
PART I: Solutions to Odd-Numbered Exercises and Practice Tests
b~
79. C=arctan-=-- =~ a=0
a2
~/a2+b2=2 ==> b=2
B=I
2 sin(0 + ~)= (0)(sin 0)+ (2)(cos 0) = 2 cos 0
81o From the figure, it appears that u + v = w. Assume that u, v, and w are all in Qua&ant I. From the figure:
s1
tan u - 3s 3
s1
2s 2
tan w = - = 1
tan v -
$
tan u + tan v
1/3 + 1/2
5/6
tan(u + v) =
=
- 1 = tan w.
1 - tan u tan v
1 - (1/3)(1/2)
1 - (1/6)
Thus, tan(u + v) = tan w. Because u, v, and w are all in Quadrant I, we have
arctan[tan(u + v)] = arctan[tan w]
U+V=W.
tan ~r + tan 0
1 - tan ~r tan 0
0 + tan 0
-1’ - (0) tan 0
=tanO
83. tanOr + O) =
6.28
cos + h - cos 6
85. f(h) =
h
g(h) = cos --" h
sin ~ \~]
(a) The domains are both (-~, 0), (0, ~x~).
(b)
h
f(h)
0.01
0.02
0.05
0.1
0.2
0.5
-0.5043 -0.5086 -0.5214 -0.5424 -0.5830 -0.6915
-0.5043 -0.5086 -0.5214 -0.5424 -0.5830 -0.6915
(c)
1
(d) The values tend to y = 2"
272
PART L" Solutions to Odd-Numbered Exercises and Practice Tests
87, x = O: y = -½(0 - 10) + 14 = 5 + 14 = 19. y-intercept: (0, 19)
1
y=O: O=-~(x-10)÷ 14=-½x + 19 ~ x= 38. x-intercept: (38,0)
89. x = o: 12(0) - 91 - 5 = 9 - 5 = 4. y-intercept: (0, 4)
y = O: 12x - 91 = 5 ==> x = 7, 2. x-intercepts: (2, 0), (7~ O)
= ~ because cos ~ = T
Section 5.5
93. arcsin 1 = _-- because sin _-- = 1.
2
2
Multiple-Angle and Product.Sum Formulas
You should know the following double-angle formulas.
(a) sin 2u = 2 sin u cos u
(b) cos 2u = COS2 U -- sinz u
= 2 cosz u - 1
= 1 - 2 sinz u
2 tan u
1 - tanz u
[] You should be able to reduce the power of a trigonometric function.
(c) tan 2u -
1 - cos 2u
1 + cos 2u
(a) sinz u =
(b) cos2 u =
2
2
[] You should be able to use the half-angle formulas.
(a) sin~=_+
u ~/1-cosu
2u ~/l+cosu
(b) cos~=+
[] You should be able to use the product-sum formulas.
1
(a) sin u sin v =~ [cos(u - v) - cos(u + v)]
1
(c) sin u cos v = ~ [sin(u + v) + sin(u - v)]
2
1 - cos 2u
(c) tanz u =
1 + cos 2u
u 1 - cos u
sin u
(c) tan sin u
2
1 + cos u
1
(b) cos u cos v = ~ [cos(u - v) + cos(u + v)]
1
(d) cos u sin v = ~ [sin(u + v) - sin(u - v)]
[] You should be able to use the sum-product formulas.
(a) sin x + sin y = 2 sln~--~)cos --~
(b) sin x- sin y = 2 cos~--~)sin ~
(c) cos x + cos y = 2 cos~--~)cos
--~
[x + y’~ (x-y)
(d) cosx-cosy=-2sinX+Y sin x-y