C H A P T E R 5 Analytic Trigonometry

64. (a) f x 2 sin x cos 2x
max: 0.5240, 1.5 min: 1.5708, 1.0
max: 2.6180, 1.5 min: 4.7124, 3.0
(b)
2 cos x 4 sin x cos x 0
2 cos x1 2 sin x 0
2 cos x 0
x 3
,
2 2
1.5708, 4.7124
65. fx tan
Since tan 4 1, x 1 is the smallest nonnegative
fixed point.
x
4
0
3
−3
1 2 sin x 0
sin x 1
2
x 5
,
6 6
Section 5.3 Solving Trigonometric Equations 467
2
0.5240, 2.6180
66. Graph y cos x and y x on the same set of axes.
Their point of intersection gives the value of c
such that fc c ⇒ cos c c.
−3 3
c 0.739
2
−2
(0.739, 0.739)
67. fx cos
(a) The domain of fx is all real numbers x except x 0.
(b) The graph has y-axis symmetry and a horizontal asymptote at y 1.
(c) As x → 0, fx oscillates between 1 and 1.
2
(d) There are infinitely many solutions in the interval 1, 1. They occur at x where n is any integer.
2n 1
(e) The greatest solution appears to occur at x 0.6366.
1
x
sin x
68. fx
x
(a) Domain: all real numbers except x 0.
(b) The graph has y- axis symmetry.
(c) As x → 0, fx → 1.
sin x
(d) 0 has four solutions in the interval 8, 8.
x
sin x 1
0 x
sin x 0
x 2, , , 2
69.
1
cos 8t 3 sin 8t 0
12
y 1
cos 8t 3 sin 8t
12
cos 8t 3 sin 8t
1
tan 8t
3
8t 0.32175 n
t 0.04 n
8
In the interval 0 ≤ t ≤ 1, t 0.04, 0.43, and 0.83.