Download Problems

426
chapter 5 Trigonometric Functions
Trigonometric identities with θ + nπ
cos(θ + nπ ) =
sin(θ + nπ ) =
⎧
⎨cos θ
if n is an even integer
⎩− cos θ
if n is an odd integer
⎧
⎨sin θ
if n is an even integer
⎩− sin θ
if n is an odd integer
tan(θ + nπ ) = tan θ
if n is an integer
The first two identities above hold for all values of θ. The third identity above
π
holds for all values of θ except the odd multiples of 2 ; these values must be
excluded because tan(θ + π ) and tan θ are not defined for such angles.
exercises
1.
For θ = 7◦ , evaluate each of the following:
(a) cos2 θ
(b) cos(θ 2 )
[Exercises 1 and 2 emphasize that cos2 θ does
not equal cos(θ 2 ).]
2.
For θ = 5 radians, evaluate each of the following:
(a) cos2 θ
3.
(b) cos(θ 2 )
For θ = 4 radians, evaluate each of the following:
(a) sin2 θ
(b) sin(θ 2 )
[Exercises 3 and 4 emphasize that sin2 θ does
not equal sin(θ 2 ).]
4.
For θ = −8◦ , evaluate each of the following:
(a) sin2 θ
(b) sin(θ 2 )
π
5. cos(− 12
)
22. cos
6. sin(− π8 )
23.
π
sin 12
π
cos 8
24.
π
sin(− 12 )
cos(− π8 )
π
tan 12
π
tan 8
π
tan(− 12
)
π
tan(− 8 )
cos 25π
12
cos 17π
8
sin 25π
12
17π
sin 8
tan 25π
12
17π
tan 8
cos 13π
12
26.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
In Exercises 5–38, find exact expressions for the
indicated quantities, given that
√
√
2+ 3
2− 2
π
π
and sin 8 =
.
cos 12 =
2
2
π
[These values for cos 12
and sin π8 will be derived
in Examples 4 and 5 in Section 6.3.]
19.
20.
21.
25.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
9π
8
sin 13π
12
sin 9π
8
13π
tan 12
9π
tan 8
cos 5π
12
cos 3π
8
5π
cos(− 12 )
cos(− 3π
)
8
5π
sin 12
3π
sin 8
sin(− 5π
)
12
3π
sin(− 8 )
tan 5π
12
tan 3π
8
tan(− 5π
)
12
3π
tan(− 8 )
Suppose u and ν are in the interval ( π2 , π), with
tan u = −2
and
tan ν = −3.
In Exercises 39–66, find exact expressions for the
indicated quantities.
section 5.6 Trigonometric Identities 427
39. tan(−u)
46. sin ν
53. tan(u + 8π )
60. tan(ν + 3π )
40. tan(−ν)
47. sin(−u)
54. tan(ν − 4π )
π
61. cos( 2 − u)
41. cos u
48. sin(−ν)
55. cos(u − 3π )
62. cos( π2 − ν)
42. cos ν
49. cos(u + 4π )
56. cos(ν + 5π )
63. sin( π2 − u)
43. cos(−u)
50. cos(ν − 6π )
57. sin(u + 5π )
64. sin( π2 − ν)
44. cos(−ν)
51. sin(u − 6π )
58. sin(ν − 7π )
65. tan( π2 − u)
45. sin u
52. sin(ν + 10π )
59. tan(u − 9π )
66. tan( π2 − ν)
problems
67. Show that
(cos θ + sin θ)2 = 1 + 2 cos θ sin θ
for every number θ.
[Expressions such as cos θ sin θ mean
(cos θ)(sin θ), not cos(θ sin θ).]
68. Show that
1 + cos x
sin x
=
1 − cos x
sin x
for every number x that is not an integer multiple of π .
69. Show that
cos3 θ + cos2 θ sin θ + cos θ sin2 θ + sin3 θ
= cos θ + sin θ
for every number θ.
[Hint: Try replacing the cos2 θ term above with
1 − sin2 θ and replacing the sin2 θ term above
with 1 − cos2 θ.]
70. Show that
tan2 θ
1 + tan2 θ
π
for all θ except odd multiples of 2 .
sin2 θ =
71. Find a formula for cos θ solely in terms of
tan θ.
72. Find a formula for tan θ solely in terms of sin θ.
73. Is cosine an even function, an odd function, or
neither?
74. Is sine an even function, an odd function, or
neither?
75. Is tangent an even function, an odd function, or
neither?
76. Explain why sin 3◦ + sin 357◦ = 0.
77. Explain why cos 85◦ + cos 95◦ = 0.
78. Pretend that you are living in the time before
calculators and computers existed, and that
you have a table showing the cosines and sines
of 1◦ , 2◦ , 3◦ , and so on, up to the cosine and
sine of 45◦ . Explain how you would find the
cosine and sine of 71◦ , which are beyond the
range of your table.
79. Suppose n is an integer. Find formulas for
sec(θ + nπ ), csc(θ + nπ ), and cot(θ + nπ )
in terms of sec θ, csc θ, and cot θ.
80. Restate all the results in boxes in the subsection on Trigonometric Identities Involving a
Multiple of π in terms of degrees instead of
in terms of radians.
81. Show that
cos(π − θ) = − cos θ
for every angle θ.
82. Show that
sin(π − θ) = sin θ
for every angle θ.
83. Show that
π
2
cos(x +
) = − sin x
for every number x.
84. Show that
sin(t +
π
2
) = cos t
for every number t.
85. Show that
tan(θ +
π
2
)=−
1
tan θ
for every angle θ that is not an integer multiπ
ple of 2 . Interpret this result in terms of the
characterization of the slopes of perpendicular
lines.