272
PART I: 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: 0 = -g(x10)÷ 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), (73 O)
91. arccos -~- = ~ because
Section 5.5
93. arcsin 1 = -~ because sin ~ = 1.
COS ~ = ~
62
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 - sin2 u
= 2 cos2 u - 1
= 1 - 2 sin2 u
(c) tan 2u -
2 tan u
1 - tan2 u
[] You should be able to reduce the power of a trigonometric function.
1 - cos2u
1 + cos 2u
(a) sinz u =
(b) cos2 u =
2
2
[] You should be able to use the half-angle formulas.
(a) sin~=+
(b) cos==+
zu~l-c°suzu~/l+c°su
2
1 - cos 2u
(c) tanz u = -1 + cos 2u
2
u 1 - cos u
sin u
(c) tan sin u
2
1 + cos u
[] You should be able to use the product-sum formulas.
1 [cos(u - v) - cos(u + v)]
(a) sin u sin v =~
1
(b) cos u cos v = ~ [cos(u - v) + cos(u + v)]
1
(c) sin u cos v = ~ [sin(u + v) + sin(u - v)]
!
(d) cos u sin v = ~ [sin(u + v) - sin(u - v)]
[] You should be able to use the sum-product formulas.
(a) sinx+siny 2sin x+y
x-y
(c) cos x + cos y = 2 cos~--~)cos --~
x+y x-y
(b) sinx-siny = 2 cos(---~)sin(--~)
(d) cosx-cosy=-2sinX+Y sin x-y
273
PART I: Solutions to Odd-Numbered Exercises and Practice Tests
Solutions to Odd-Numbered Exercises
Figure for Exercises 1-7
sin 0 = ~
COS 0 = ~
tan 0 = ~
3. cos 20=2cos20- 1
= 2(~)~- 1
1. sin 0 = ~
32 25
-- 25 25
7
-- 25
5. tan20 --.
2 tan 0
1 - tan2 0
2(3/4)
1 - (3/4)2
3/2
1 - (9/16)
3 16
27
24
9. Solutions: O, 1.047, 3.142, 5.236
sin 2x - sin x = 0
2 sin x cos x - sin x = 0
sin x(2 cos x - 1) = 0
sin3c=O or 2cosx- 1 =0
1
X-" O,’rr
cosx =-2
¯ r 5~r
x = O, ~, "n’, 3
11. Solutions: 0.1263, 1.4445, 3.2679, 4.5860
1
sin 20
1
--.
.2sin0cos0
1
..._
2(3/5)(4/5)
25
24
7. csc 20 -
274
PART I: Solutions to Odd-Numbered Exercises and Practice Tests
13. Solutions: 1.047, 3.142, 5.236
15. Solutions: O, 1.571, 3.142, 4.712
sin 4x = -2 sin 2x
sin4x + 2 sin2x = 0
cos 2x= -cos x
2x2cos
1 =-cosx
2 sin 2x cos 2x + 2 sin 2x = 0
2 sin 2x(cos 2x + 1) = 0
2cos2x+cosx- 1=0
(2 cos x - 1)(cos x + 1) = 0
2cosx= 1 or cosx=-I
1
2
COS X = --
2sin2x=O
or
cos2x+ 1=0
sin 2x = 0
X=~
2x = nqr
n
x = ~’/r
,tr 5"tr
3’3
cos 2x = -1
x = -~ + nq~"
","
3"n"
x=O,-~, ,rr, 2
17. 8 sin x cos x = 4(2 sin x cos x) = 4 sin 2x
-tr 3,0"
x-2’ 2
19. 5 - 10 sin2 x = 5(1 - 2 sin2 x) = 5 cos 2x
4
3
¢r
--- n
21. sinu = ~. O<u<-~ :=* cosu
5
3 4 24
5 5 25
16 9 7
- -cos 2u = cos2 u - sin2 u 25 25 25
sin2u=2sinucosu=2" ¯ -
2 tan u
2(3/4) 24
tan 2u 1 - tan2u 1 - (9/16) 7
tan 2u -
2x = ¢r + 2n~r
7r
2 tan u
2(1/2) 4
1 - tan2 u 1 - (1/4) 3
.-..
--.
1 + 2 cos 2x + cos2 2x
4
1 + 2 cos 2x + (1 + cos 4x)/2
4
2 + 4 cos 2x+ 1 + cos 4x
8
3 + 4 cos 2x + cos 4x
8
= ~(3 + .4 cos 2x + cos 4x)
275
PART I: Solutions to Odd-Numbered Exercises and Practice Tests
1 - cos~ 2x
4
= 1 - 1 + cos
2
= ~(2 - 1 - cos 4x)
= -~(1 - cos 4x)
¯
29° sin2 x cos4 x = sin2 x COS2 X gOS2 X = ’
_! (1 - cos 2x)(1 + cos 2x)(1 + cos 2x)
8
_! (1 - cos2 2x)(1 + cos 2x)
8
_! (1 + cos 2x - cos2 2x - cos3 2x)
8
_1
8 [l+cos2x- (l+cos4x)2
- cos2x(l+c°s4x)]2
__ 1 [2 + 2 cos 2x - 1 - cos 4x - COS 2x - cos 2x cos 4x]
16
cos 2x + ~ cos 6x
= 1-~ 1 + cos 2x - cos 4x 1
(2 + 2 cos 2x - 2 cos 4x - cos 2x - cos 6x)
32
__ 1 (2 + cos 2x - 2 cos 4x - cos 6x)
32
Figure for Exercises 31 - 35
sin 0 = ~
cos 0 - 13
0 /1 + cos 0 ~/i + (12/13) _
231. cos ~ =~/
2
0
sin 0
5/13
51
33. tan ....
2 1 + cos 0 1 + (12/13) 25 5
0 1
1
35. CSC--=~=
2 sin(O/2) ~/(1 - cos 0)/2
2~22~ 5 5~,/~
- ~ 26
0o
.5
37. 2 s]n ~ cos ~ = sin O = 1"~"
cos
2
39. sin 15°=sin~-30° =
cos 15°= cos ~" 30°2
_
-
2
=
-_
- 2w
2
1
°
sin 30
2
1
=~=2
tan 15° tan(21-. 300)=
°
1+ cos30 1 + _.~_~
2
+~
=
-
41. sin 112°30’= sin(~. 225o)=
cos
2
cos 112° 30’ = cos . 225° = _
2
-~/-~/2
= sin225°
tan 112° 30’= tan(~. 2250)
°- -1-(-,/~12) =-11+cos225
43. sln~ =sm.=
N!/~+ cos(or/4)
= ~/2 + ~
2
COS "ff -- COS
tan ~ = tan
=
sin(or/4) _ ~/2 = ~- 1
1 + cos(w/4) - 1 + (v/~/2)
3¢r (½ ~)~//1 - cos (3~’/4)
45. siny=sin "-- =
2
=
3,n"
=
cos-if- = cos (~¯ ~ff)
~/~ + cos3"rr/4
2
=
tan-if- = tan
--- =
sin -4
2
3~-1 + cos -:-. 1
4
5"
12
47. sin u = -- ~<u<-rr :=~ cosu =
13’2
sin(~)=~/1-c°su- N/1+(12/13)2
4~=
2
13
2 - 5V/~~26
(~)=N//~ + cos u ~/1- (12/13)
cos
2 2
- 26
tan - 1 + cos u 1 - (12/13) 1
277
PART I: Solutions to Odd-Numbered Exercises and Practiee Tests
8. 3’rr
~ ~5<
49. tan u =
~ ~u < 2~, Quadrant IV
2
sin u = -~
COS /~ =
-- COS U
2
5
= ~1 - (5/,,/~)2
.-,,.
~/ 2~/~~
178
cos ~ ==-(5/-,/~)/-,/~9 +__25
(u) ~/1 + cosu ~/12 +
5
tan
sin u
-8
178
=-’N/ 2-v/’~ =-
8
51. N/1 - cos2 6x_- Isin 3x[
,/(1 - cos 8x)/2
~/(1 + cos 8x)/2
53. - ~/~ - cos 8x
+ cos 8x
= _[ sin 4x]
Icos 4xl
x
55.
57.
sinT-cosx=O
2
~/1 - cos x = COS X
2
1 - cos x
-- COS2 X
2
O=2cosZx+cosx- 1
= (2 cos x - 1)(cos x + 1)
1
cosx=or cosx=-I
2
,n" 5"n"
X-X -- "B"
3’ 3
By checking these values in the original equations,
we see that x = "n’/3 and x = 5 ¢r/3 are the only
solutions, x = -n" is extraneous.
+
x
cos~ - sinx = 0
+- ~/~+ cos x - sin x
2
1 + cos x
- sin2 x
2
1 + cosx = 2sinzx
1 +cosx=2-2cosZx
2x+cosx2cos
1 =0
(2 cosx- 1)(cosx+ 1)=0
2cosx- 1 =0
or cosx+ 1=0
1
2
"rr 5"tr
X-3’ 3
,tr 5,n"
x=~, ¢r, 3
COS X = --
cos x = - 1
qr/3, ¢r, and 5qr/3 are all solutions to the equation.
sin ~ cos ~ = 6 ¯
sin +
+ sin -
= 3[sin~+ sinO]=3 sin2Cr3
278
PART 1: Solutions to Odd-Numbered Exercises and Practice Tests
6]. sin 5Ocos 30 = ½[sin (50 + 30) + sin(50 - 30) = ½(sin 8O + sin 20)
63. 5 cos(-5/~) co~ 3/~ = 5 ¯ ½[cos(-5~ - 3/~) + cos(-5/~ + 3~)] = ~[cos(-8~) + cos(-2/~)]
= ~(cos 8~ + cos 2~)
\
65. sin 60° + sin 30°=2 sin|60° +230°.!/\ cos|60°30°-1 = 2{sin ,*5° cos15°
2
\
/ \
/
67. sin 5 0 - sin 0 = 2 cos
(5o2+~)
= 2 cos 30. sin 20
= - 2 sin 0 sin 7 = - 2sin 0
2
73.
75.
sin 6x + sin 2x = 0
2
cos 2x
-1=0
sin 3x - sin x
cos 2x
=1
sin 3x - sin x
2
sin/6X/ 2
sin 4x cos 2x = 0
sin4x=O
cos 2x
=1
2 cos 2x sin x
or cos2x=O
2 sin x = 1
/
4x = n~"
X--
n’n"
4
2x’= ~ + n,rr
1
sin x = 2
,n" n’n"
42
X=--+J
¢r
x-6,6
2
2
0.28
0 ~ 6,28
-2
In the interval we have
,r ¢r 3~r 5¢r 3¢r
x=O’ 4’ 2’ 4’ ¢r’ 4’ 2’ 4
25
(~3)2
- 169
77. sin2 a =
sin2a= 1-cos2a= 1 --(1~)2 = 1
lz14 25
169 -- 169
279
PART I: Solutions to Odd-Numbered Exercises and Practice Tests
=
79. sin a cos fl =
sin a cos fl = cos -
,
sin -/3 =
=
1
sin 20
1
2 sin 0 cos 0
1 1
sin 0 2 cos 0
csc 0
2 cos 0
81. csc20 -
83. cos2 2a - sin: 2or = cos[2(2ot)]
= cos 4a
85. (sin x + cos x)2 = sin:~ x + 2 sinx cos x + cos:~ x
= (sin: x + cos2 x) + 2 sin x cos x
= 1 + sin 2x
u 1
87. sec- = ±--
89. cos 3/3 = cos(2/3 +/3)
cos (u/2)
= cos 2/3 cos/3 - sin 2/3 sin/3
= (cos2/3 - sin2/3) cos/3 - 2 sin/3 cos/3 sin/3
-..
= cos3/3 - sin2 fl cos/3 - 2 sin2/3 cos/3
= cos3/3 - 3 sin2/3 cos/3
/ 2 sin u
+-~/-sin u(1 + cos u)
/_ 2 sin u
+-~/-sin u + sin u cos u
/.
(2 sin u)/(cos u)
- ~/(sin u)/(cos u) + (sin u cos u)/(cos u)
+
= +./.: 2_ta__n_u
- ~/tanu + sinu
91.
cos 4x - cos 2x
2 sin 3x
2 sin 3x
..._
-2 sin 3x sin x
2 sin 3x
= - sin x
1 - cos 2x 1
93. sinz x =
2
2
2
cos 2x
2
28O
PART I: Solutions to Odd-Numbered Exercises and Practice Tests
x
2cos~ - sinx = 0
(b)
95. (a) y=4sin~+cosx
2+
4
2
4(1+2 cos X) = sin2 x
2(1 +cosx)= 1-cos2x
~ 8.28
0
0
cos2x+2cosx+ 1 =0
Maximum: (or, 3)
(cosx + 1)2 = 0
COS X = -- 1
x
97. (a) y = cos~ + sin 2x
x
(b) 2 cos 2x - sin ~ = 0 has 4 zeros on [0, 2¢r). Two of
ezeros are x = 0.699 and x = 5.584. (The other
4
two are 2.608, 3.675)
Maximum: (0.699, 2.864)
Minimum: (5.584, - 2.864)
99. sin(2 arcsin x) = 2 sin(arcsin x) cos(arcsin x) = 2xv/~ - x2
1
101. cos(2 arcsin x) = 1 - 2 sin2(arcsin x)
= 1 - 2x2~
103. r = ~ 1vo sin 20
- ± Vo(2sin 0 cos 0)
= N1vo sin 0 cos 0
~
01
105. sin~ =
0
sin 2
0
2
107. False. If x = or, sin ~ = sin ~ = 1, whereas
1
4.5
1
arcsin ~-~ 0.2241
0 ~ 0.4482 or 25.68°
-- COS "n"
2
= --1
281
PART I: Solutions to Odd-Numbered Exercises and Practice Tests
109. f(x) = 2 sin x 2 cos
=1
2
(a)
~
(b) The graph appears to be that of y = sin 2x.
(c) 2 sinxI2 cosZ(~)-l]=2 sinx[21+ cosx
= 2 sin x [cos x]
_~
= sin 2x
~~
111o (a) Complement: 2 18
8,tr 4~r
18 - 9
,tr 17~r
18 18
Supplement: ~r
113. f(x) = -~5 sin ~
y
Period: 4,rr
~ 9~r
(b) Complement: 2 20
~r
2"-~
9~r= 2"--~ll~r
Supplement: ~r- 2--~
1
7rx
115. f(x) = ~ sec T
y
Period: -" = 4
5
Amplitude: ~
2
Review Exercises for Chapter 5
Solutions to Odd-Numbered Exercises
Cos x
~ = cot x
sin x
7. csc
9. sec(--x) "- sec x
3
4
11. sin x = ~, cos x = if, Quadrant I
3
sin x 5 3
tan x cos x 4
5
4
cot x = 3
5
3
CSC X = --
5
4
sec X -" --
sin x = --~ =
4
tan x = - 1
cot x = - 1
SeC X = -~
CSC X = -- ~
~ .. Quadrant IV
= sec x