Download 7.1 Exercises

SECTION 7.1
7.1
1. cos t tan t
2. cos t csc t
3. sin u sec u
4. tan u csc u
5. tan2x sec2x
6.
7. sin u cot u cos u
sec u cos u
sin u
11–24
■
sec x
csc x
8. cos2u 11 tan2u 2
10.
cot u
csc u sin u
1 cos y
13.
1 sec y
tan x
14.
sec1x 2
sec2x 1
15.
sec2x
sec x cos x
16.
tan x
1 csc x
17.
cos x cot x
sin x
cos x
18.
csc x
sec x
3
20. tan x cos x csc x
21.
2 tan x
1
sec2x
22.
2
1 cot A
csc A
23. tan u cos1u 2 tan1u2
27.
1
csc2b
cos x
sin x
1
sec x
csc x
1sin x cos x 2 2
sin2x cos2x
sin2x cos2x
1sin x cos x 2 2
40. 1sin x cos x 2 4 11 2 sin x cos x2 2
41.
sec t cos t
sin2t
sec t
42.
1 sin x
1sec x tan x 2 2
1 sin x
43.
1
1 tan2y
1 sin2y
44. csc x sin x cos x cot x
45. 1cot x csc x 2 1cos x 12 sin x
46. sin4u cos4u sin2u cos2u
47. 11 cos2x 2 11 cot2x 2 1
48. cos2x sin2x 2 cos2x 1
49. 2 cos2x 1 1 2 sin2x
1 sin u
cos u
cos u
1 sin u
25.
39.
2
19.
50. 1tan y cot y 2 sin y cos y 1
51.
sin a
1 cos a
sin a
1 cos a
52. sin2a cos2a tan2a sec2a
53. tan2u sin2u tan2u sin2u
cos x
sec x tan x
■
37. 11 cos b 2 11 cos b 2 Simplify the trigonometric expression.
12. cos x sin x cos x
25–88
36. 1sin x cos x 2 2 1 2 sin x cos x
38.
sin x sec x
11.
tan x
24.
533
Exercises
1–10 ■ Write the trigonometric expression in terms of sine and
cosine, and then simplify.
9.
Trigonometric Identities
54. cot2u cos2u cot2u cos2u
55.
Verify the identity.
sin u
cos u
tan u
26.
cos u sec u
cot u
tan u
28.
tan x
sin x
sec x
cot x sec x
1
csc x
57.
sin x 1
cos2x
tan „
sin „
56.
sin x 1
sin „ cos „
1 tan „
1sin x 12 2
1sin t cos t 2 2
sin t cos t
2 sec t csc t
58. sec t csc t 1tan t cot t 2 sec2t csc2t
59.
1
1 tan2u
1 tan2u
cos2u sin2u
31. sin B cos B cot B csc B
60.
1 sec2x
1 cos2x
1 tan2x
33. cot1a 2 cos1a2 sin1a 2 csc a
61.
sec x
sec x 1sec x tan x 2
sec x tan x
35. tan u cot u sec u csc u
62.
sec x csc x
sin x cos x
tan x cot x
tan y
sec y cos y
29.
csc y
cos √
csc √ sin √
30.
sec √ sin √
32. cos1x 2 sin1x 2 cos x sin x
34. csc x 3 csc x sin1x2 4 cot2x
534
CHAPTER 7
63. sec √ tan √ 64.
65.
Analytic Trigonometry
89–94 ■ Make the indicated trigonometric substitution in
the given algebraic expression and simplify (see Example 7).
Assume 0 u p/2.
1
sec √ tan √
sin A
cot A csc A
1 cos A
89.
sin x cos x
sin x cos x
sec x csc x
sin x
1 cos x
66.
2 csc x
sin x
1 cos x
67.
csc x cot x
cot x
sec x 1
csc x cot x
cos2x
sec2x
2
68.
90. 21 x 2, x tan u
91. 2x 2 1, x sec u
92.
93. 29 x 2, x 3 sin u
94.
1
x 2 24 x 2
,
2x 2 25
,
x
x 2 tan u
x 5 sec u
95–98 ■ Graph f and g in the same viewing rectangle. Do the
graphs suggest that the equation f 1x2 g1x 2 is an identity?
Prove your answer.
96. f 1x 2 tan x 11 sin x 2, g1x2 97. f 1x2 1sin x cos x 2 2,
cos u
sec u tan u
1 sin u
sin x cos x
1 sin x
g1x2 1
98. f 1x 2 cos x sin x, g1x 2 2 cos2x 1
4
sin u csc u
cos u
73.
1 sin u
cos u cot u
4
99. Show that the equation is not an identity.
(a) sin 2x 2 sin x
(b) sin1x y 2 sin x sin y
(c) sec2x csc2x 1
1
(d)
csc x sec x
sin x cos x
74.
cos x sin x
1 tan x
1 tan x
cos x sin x
75.
cos2t tan2t 1
tan2t
sin2t
76.
1
1
2 sec x tan x
1 sin x
1 sin x
1
1
77.
2 sec x
sec x tan x
sec x tan x
78.
, x sin u
95. f 1x2 cos2x sin2x, g1x 2 1 2 sin2x
tan √ sin √
tan √ sin √
tan √ sin √
tan √ sin √
71. sec4x tan4x sec2x tan2x
72.
21 x 2
2
69. tan2u sin2u tan2u sin2u
70.
x
1 sin x
1 sin x
4 tan x sec x
1 sin x
1 sin x
Discovery • Discussion
100. Cofunction Identities In the right triangle shown,
explain why y 1p/2 2 u. Explain how you can
obtain all six cofunction identities from this triangle, for
0 u p/2.
√
79. 1tan x cot x 2 2 sec2x csc2x
80. tan2x cot2x sec2x csc2x
81.
sec u 1
1 cos u
sec u 1
1 cos u
83.
sin3x cos3x
1 sin x cos x
sin x cos x
84.
tan √ cot √
sin √ cos √
tan2√ cot2√
82.
85.
1 sin x
1tan x sec x 2 2
1 sin x
86.
tan x tan y
tan x tan y
cot x cot y
87. 1tan x cot x 2 4 csc4x sec4x
1 tan x
cot x 1
cot x 1
1 tan x
88. 1sin a tan a 2 1cos a cot a 2 1cos a 12 1sin a 12
u
101. Graphs and Identities Suppose you graph two functions, f and g, on a graphing device, and their graphs
appear identical in the viewing rectangle. Does this prove
that the equation f 1x 2 g1x 2 is an identity? Explain.
102. Making Up Your Own Identity If you start with a
trigonometric expression and rewrite it or simplify it,
then setting the original expression equal to the rewritten
expression yields a trigonometric identity. For instance,
from Example 1 we get the identity
cos t tan t sin t sec t
Use this technique to make up your own identity, then give
it to a classmate to verify.