256
PART I: Solutions to Odd-Numbered Exercises and Practice Tests
89. f(x) = 5-x - 2
y
4
-2
91. s = rO
s 26
0 ....
r 11
95. Qua&ant III
93. Qua&ant III
~ 2.3636 radians
Seclion 5.3
Solving Trigonometric Equations
[] You should be able to identify and solve trigonometric equations.
[] A trigonometric equation is a conditional equation. It is true for a specific set of values.
[] To solve trigonometric equations, use algebraic techniques such as collecting like terms, taking square roots,
factoring, squaring, converting to quadratic form, using formulas, and using inverse functions. Study the
examples in this section.
[] Use your graphing utility to calculate solutions and verify results.
Solutions to Odd-Numbered Exercises
1. 2cosx- 1=0
3. 3tan22x- 1=0
(a) 3 tan \-~2] - 1 = 3 ~tn2-~- 1
(b) 2 cos--2-- 1 = 2 - 1 ---0
=3
-1
=0
(b) 3Itan (10~r/q2\’~’-]1 - 1 = 3 tan2~-~- 1
__
=0
257
PART I: Solutions to Odd-Numbered Exercises and Practice Tests
5. 2cos2x+ 3cosx+ 1=0
2/’4"rr’~
-3 + 1 =0
(a) 2cos ~-~-) +3cos T + 1 = 2 (4"rr)
(_~)2 (_~)
(b) 2cos2"rr+ 3cos~+ 1 = 2(-1)2- 3 + 1 = 0
J
9. y = tan t--~--~ - 3
From the graph in the textbook we see that
the curve has x-intercepts at x = +2.
7. y = sin-~-+ 1
From the graph in the textbook we see that
the curve has x-intercepts at x = -1 and at
11. 2cosx + 1 = 0
2 cos x = -1
13. ~secx-2=0
.!g sec x = 2
1
2
2¢r
X "-- ~
3
4qr
orx = ~
3
COS X -- ----
sec x cos x --
15. 3csc2x-4=O
4
3
CSC2 X ----
2
,/~
2
sinx= ±
ll~r
x = =orx = ~
6
6
X--
2
"rr 2~" 4"rr 5~r
3’3’3’3
17. 2sin22x= 1
1 ./2
sin2x=+-~=± 2
2x =3"n"T,z =5"trT,2x =7"rr
T,
9’rr
11o
13rr
15~"
2x=~---, 2x=---~---,2x- 4 , 2x= 4
,rr 3,n" 5qr 7~r 9"rr ll,tr.13¢r 15"n"
Thus, X -(8 solutions).
8’8’8’8’8’8’8’8
19. 4cos2x- 3 =0
3
4
COS2 X : --
COS X ---- i
2
,rr 5"rr 7’rr ll~r
6’6’6’ 6
21.
sin2 x = 3 cos2 x
sin2x - 3(1 - sin2x) = 0
4 sin2 x = 3
sinx= ±
X--
2
"rr 2"rr 4~r 5"rr
3’3’3’3
258
PART I: Solutions to Odd-Numbered Exercises and Practice Tests
23. (3 tan2 x - 1)(tan2 x - 3) =0
3tan2x- 1=0
or
tan2 x - 3 = 0
1
tanx = +~r~
tan x = + V’~
~r 7~
6’6
5~r ll,rr
orx= 6’ 6
X "-" ~ ~
or X --
COS3X "- COSX
25.
"n" 4,rr
3’3
2"n" 5,rr
3’3
X ~- ~ ~
27.
COS3X -- COSX -- 0
cos x(cos~ x - 1) - 0
cosx=O
or cos2x- 1 =0
qr 3~r
COSX = +__1
x-2,2
3 tanax - tan x = 0
tan x(3 tanz x - 1) = 0
tanx=O
or 3tan2x- 1=0
x = O, ,rr
x=O,
29.
33.
sec2x - secx - 2 = 0
(sec x - 2)(sec x + 1) = 0
sec x - 2 = 0
or sec x + 1 = 0
sec x = 2
SeC X -- -- 1
"n" 5,0"
3’3
31. 2sinx+cscx=O
2 sin x +1 ~ = 0
sin x
2sin2x+ 1=0
Since 2 sin2 x + 1 > O, there are no solutions.
cscx+cotx= 1
1
cos x
sin x sin x
1 +cosx=sinx
(1 + cos x)2 = sin2 x
1 +2cosx+cosZx= 1-cos2x
2cos2x + 2cosx = 0
2 cos x(cos x + 1) = 0
cosx=O
or
cosx=-I
qr
2’ 2
(3¢r/2 is extraneous.) (,r is extraneous.)
x = ¢r/2 is the only solution.
35. cos - 2
x~
24
37.
+ 2n~r
x =~ +4nqr
tan x = +~/3
3
¢r 5~" 7"¢ 11~"
x-6, 6, 6,6
1 + cos x
-0
1 - cos x
1 + cos x = 0
cosx - -1
259
39.
PART I: Solutions to Odd-Numbered Exercises and Practice Tests
2 secax + tanax - 3 = 0
2(tan2x + 1) + tan2x - 3 =0
3tan~x- 1-0
tan x = ~
3
¯r 5¢r 7~r 117r
X-"
6’6’6’ 6
41.
sec2x q- tan x = 3
43. y=9cosx- 1
x ~ 4.8237, 1.4595
¯ (1 +tan~x)+tanx=3
tan2x + tanx -- 2 = 0
6.28
(tan x + 2)(tan x - 1) = 0
tanx=--2 or tanx= 1
45. y=4sin2x-2cosx- 1
-IO
47. 4 sin3 x + 2 sin2 x - 2 sin x - 1 = 0
Graph y = 4 sinax + 2 sin2x - 2 sin x - 1.
x ~ 0.8614, 5.4218
4
o ~ 6.28
6 28
By altering the y-range to Ymin = -.5 and
Ymax = .5, you see that there are 6 solutions:
0.7854, 2.3562, 3.6652, 3.9270, 5.4978, 5.7596’.
49.
cos x cot x
1 - sin x
Graph y =
51. xcosx- 1 =0
-3
COS X
(1 - sin x) tan x
-3.
3
x ~ 4.9172
The solutions are approximately
x ~ 0.5236, x ~- 2.6180
260
PART I: Solutions to Odd-Numbered Exercises and Practice Tests
53. secZx + 0.5 tanx - 1 = 0
Graph Yl - (cos x)2 + 0.5 tan x - 1.
lO
0
6.28
-2
The x-intercepts occur at x = 0, x ~ 2.6779, x ~ 3.1416 and x ~ 5.8195.
55.
12sinzx- 13sinx+3 =0
Graph Yl = 12 sin:~ x - 13 sin x + 3.
40
-4
The x-intercepts occur at x ~ 0.3398, x ~ 0.8481, x ~" 2.2935, and x ~-- 2.8018.
57. 3tan2x+5tanx-4=0, -~,
tan x =
59. 4cos2x-2sinx+l=0,[-~,~l
4(1 -sin2x)-2sinx+ 1 =0
- 5 + ./25 - 4(- 4)(3)
2(3)
-4sinZx-2sinx+5=0
sin x =
x ~ - 1.154, 0.535
2 + ,//4- 4(-4)(5)
2(- 4)
2 _+ ./g~
-8
--
-4
x ~ 1.110
x
0
1
2
f(x)
Undef.
0.83
-1.36
61. (a)
3
4
-2.93 -4.46
5
6
-6.34 - 13.02
The zero is in the interval (1, 2) since f changes signs in the interval.
(b)
2
The interval is the same as part (a).
(c) 1.3065
~3. (a)
x
0
1
2
3
4
f(x)
-1
1.39
1.65
-0.70
- 1.94
5
6
- 1.48
The zeros are in the intervals (0, 1) and (2, 3) sincefchanges signs in these intervals.
8
The intervals are the same as part (a).
(c) 0.4271, 2.7145
65. (a) f(x) = sin x + cos x
2
(b) cosx-sinx=O
cos x = sin x
sin x
1cos X
tan x = I
¢r 57r
x-4,4
f
=sin~+cos4- 2
2
22
Therefore, the maximum point in the interval [0, 2"n’)is (,n’/4, ~ and the minimum point is (5"n’/4, --,/~).
67. (a) f(x) = x sin 2x
Maximum: (3.989, 3.958)
Minimum: (5.543,-5.520)
(b) 2x cos 2x + sin 2x = 0
y = 2x cos 2x + sin 2x
lO
6 28
0 ~~~ 6.28
5 solutions: 0, 1.014, 2.457, 3.989, 5.543.
The fourth and fifth correspond to the maximum and minimum found in part (a).
262
PART I: Solutions to Odd-Numbered Exercises and Practice Tekts
69. f(x) = tan 4 tan 0 = 0~ but 0 is not positive. By graphing y = tan-4- - x, you see that the smallest positive
fixed point is x = 1.
1
71. f(x) = cosx
(a)
(b)
(c)
(d)
(e)
The domain off(x) is all real numbers except 0.
The graph has y-axis symmetry and a horizontal asymptote at y = 1.
As x ----> 0, f(x) oscillates between - 1 and 1.
There are an infinite number of solutions in the interval [- 1, 1 ].
The greatest solution appears to occur at x -~- 0.6366.
1
y = -j~(cos 8t - 3 sin 8t)
73.
8t - 3 sin 8t) = 0
~2(cos
75. D=31 sm(3--~t - 1,4)
40
cos 8t -- 3 sin 8t
1
- = tan 8t
3
8t = 0.32175 + n’nt = 0.04 + -8
o I~1365
-40
D > 20° for 123 < t < 223 days
In the interval 0 < t _< 1, t = 0.04, 0.43, and 0.83.
77. Range = 1000 yards = 3000 feet
vo = 1200 feet per second
r = ~ Vo2 sin 20
79. Yl = 1.56e-°’22t COS 4.9t intersects Y2 = -- 1
at t ~ 1.96
3000 = (12oo)2 sin 20
sin 20 = 0.066667
20 ~-- 3.8226°
0 ~ 1.9113°
The displacement does not exceed one inch from
equilibrium after t= 1.96 seconds.
Sl. f(x)= 3 sin(O.6x- 2)
(a) Zero: sin(O.6x- 2) = 0
0.6x- 2 = 0
0.6x= 2
2 10
0.6 3
(c) -0.45xz + 5.52x - 13.70 = 0
X=
- 5.52 + ./(5.52)2 - 4(-0.45)(- 13.70)
2(- 0.45)
x = 3.46, 8.81
The zero of g on [0, 6] is 3.46. The zero is close
to the zero ~ ~ 3.33 off.
(b) g(x) = -0.45x2 + 5.52x - 13.70
5
For 3.5 < x < 6 the approximation appears to be
good.
PART I: Solutions to Odd-Numbered Exercises and Practice Tests
83. False. There might not be
periodicity, as in the equation
sin(x2) = 0
89. tan 30°-
14
x
--~ x =
2.164 radians
14 14
= --~ = 24.249
tan 30°
3
0.007 radians
16
16
91. sin 40°= ~ ==~ x =
x
sin 40°
24.892
95, f(x) = ~cot y
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 7- sin u sin v
tan u + tan v
tan(u + v) =
17- 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 ~ sin ~-
"rr
,rr ./~+1
(b) cos~+cos-~=T s
-,J’~" + 1
2