7-2 Verifying Trigonometric Identities Pages 433–436

1
2
63. a ! 12.0, B ! 70°, b ! 32.9
64. "2, " $ , 2
65. "4, 0.5
66. continuous
67. (2, "5, "3)
68. y ! " $ x # $
2
9
28
9
69. C
7-2 Verifying Trigonometric Identities
Pages 433–436
1. Answers will vary.
2. Sample answer: Squaring each
side can turn two unequal
quantities into equal quantities.
For example, "1 & 1, but
("1)2 ! 12.
4. Answers will vary.
3. Sample answer: They are the
trigonometric functions with
which most people are most
familiar.
5. cos x
cos x
!
cot x
$
csc x
1
cos x
! $$
tan x # sec x
sin x # 1
6. $$
cos x
$$
sin x
! $
1
$$
sin x
cos x
1
$$
$$
!
sin x
1
sin x + 1
$$ # $ $
cos x
cos x
1
$$
sin x # 1 ! $$
sin x # 1
$$
cos x
cos x ! $
1
cos x
cos x
cos x
$$ ! $$
sin x # 1
sin x # 1
cos x ! cosx
7. csc ! " cot !
!
csc ! " cot !
!
1
$$
csc ! # cot !
1
csc ! " cot !
$$ % $$
csc ! # cot ! csc ! " cot !
csc ! " cot !
csc ! " cot ! ! $$
csc2 ! " cot2 !
csc ! " cot !
!
csc ! " cot !
$$$
(1 # cot2 !) " cot2 !
csc ! " cot !
csc ! " cot ! ! $$
1
csc ! " cot ! ! csc ! " cot !
Glencoe/McGraw-Hill
cos x
177
8. sin ! tan !
!
sin ! tan !
!
sin ! tan !
!
sin ! tan !
!
sin ! tan !
!
sin ! tan !
!
sec ! " cos !
1
$ " cos !
cos !
1
cos2 !
$ " $
cos !
cos !
2
1 " cos !
$$
cos !
sin2 !
$
cos !
sin !
sin ! $
cos !
sin ! tan ! ! sin ! tan !
Advanced Mathematical Concepts
Chapter 7
1
4
9.
(sinA " cosA)2 ! 1 " 2 sin2A cotA
sin2A " 2 sinA cosA # cos2A
! 1 " 2 sin2A cotA
1 " 2 sinA cosA ! 1 " 2 sin2A cotA
10. Sample answer: sinx ! $
sinA
sinA
1 " 2 sinA cosA $
!
1 " 2 sin2A cotA
cosA
sinA
1 " 2 sin2A $
1 " 2 sin2A cotA
1 " 2 sin2A cotA ! 1 " 2 sin2A cotA
11. Sample answer: cos x ! "1
!
I cos !
I cot !
! $
R
R 2csc !
12. $
2
cos
I $$
sin
I cos !
$
! $
2
1
R
R 2 $$
sin !
cos !
I $$
sin !
I cos !
$
! $
1
R2
R 2 $$
sin !
13. tan A
!
I cos !
I cos !
$
! $
2
R
R2
sec A
$
csc A
tan A
1
$$
cos A
! $
1
$$
sin A
tan A
!
sin !
sin !
%$
14. cos !
!
sin ! cot !
cos !
!
sin ! $
cos !
sin !
cos ! ! cos !
sin A
$
cos A
tan A ! tan A
Glencoe/McGraw-Hill
178
Advanced Mathematical Concepts
Chapter 7
1 " sin x
$$
cos x
1
sin x
sec x " tan x ! $ " $
cos x
cos x
15. sec x " tan x
16.
!
1 # tan x
$$ ! sec x
sin x # cos x
sin x
1 # $$
cos x
$$ ! sec x
sin x # cos x
sec x " tan x ! sec x " tan x
sin x
1 # $$
cos x
cos x
$ % $$ ! sec x
cos x sin x # cos x
cos x + sin x
$$$ ! sec x
cos x(sin x + cos x)
1
$ ! sec x
cos x
sec x = sec x
17. sec x csc x
!
sec x csc x
!
sec x csc x
sec x csc x
sec x csc x
sec x csc x
sec x csc x
tan x # cot x
sin x
cos x
$ # $
cos x
sin x
sin x
sin x
! $ % $ #
cos x
sin x
cos x
cos x
$ % $
sin x
cos x
2
sin x
! $$ #
cos x sin x
cos2 x
$$
sin x cos x
2
sin x # cos2 x
! $$
cos x sin x
1
! $$
cos x sin x
1
1
! $ % $
cos x
sin x
18. sin ! # cos !
!
sin ! # cos !
!
2sin2! " 1
$$
sin ! " cos !
2sin2 ! " (sin2 ! # cos2 !)
$$$
sin ! " cos !
sin2 ! " cos2 !
sin ! # cos ! ! $$
sin ! " cos !
sin ! # cos !
!
(sin ! " cos !)(sin ! # cos !)
$$$$
sin ! " cos !
sin ! # cos ! ! sin ! # cos !
sec x csc x ! sec x csc x
Glencoe/McGraw-Hill
179
Advanced Mathematical Concepts
Chapter 7
19.
(sin A # cos A)2
!
(sin A # cos A)2
!
sec A csc A
sec A csc A
2 # $$
2
sec A csc A
$$ # $$
sec A csc A
sec A csc A
(sin A # cos A)2
!
1
sec A
1
csc A
2$ % $ # 1
(sin A # cos A)2 ! 2 cos A sin A # 1
(sin A # cos A)2 !
2 cos A sin A # sin2 A # cos2 A
(sin A # cos A)2 ! (sin A # cos A)2
cos y
1 # sin y
$$ ! $$
21.
1 " sin y
cos y
cos y
1 # sin y
1 # sin y
$$ % $$ ! $$
1 " sin y
1 # sin y
cos y
cos y(1 # sin y)
1 # sin y
$$
! $$
2
1 " sin y
cos y
cos y(1 # sin y)
1 # sin y
$$
! $$
2
cos y
cos y
1 # sin y
1 # sin y
$$ ! $$
cos y
cos y
2
cot x
23. csc x " 1 ! $$
csc x # 1
csc2 x " 1
csc x " 1 ! $$
csc x # 1
(csc x # 1)(csc x " 1)
csc x " 1 ! $$$
csc x # 1
csc x " 1 ! csc x "1
20.
(sin ! " 1)(tan ! # sec !) ! "cos !
sin ! tan ! " tan ! #
sin ! sec ! " sec ! ! "cos !
sin!
cos!
sin!
cos!
1
1
sin ! $ " $
cos !
cos !
2
sin ! " sin ! # sin ! " 1
$$$
cos !
sin2 ! " 1
$$
cos !
"cos2 !
$
cos !
sin ! $ " $ #
!
"cos !
!
"cos !
!
"cos !
!
"cos !
"cos ! ! "cos !
22.
cos ! cos("!) " sin ! sin("!) ! 1
cos ! cos ! " sin !("sin !) ! 1
cos2 ! # sin2 ! ! 1
1!1
24. cosB cot B
!
cosB cot B
!
cosB cot B
!
cosB cot B
!
cosB cot B
!
cosB cot B
!
csc B " sin B
1
$ " sin B
sin B
1
sin2 B
$ " $
sin B
sin B
2
1 " sin B
$$
sin B
cos2 B
$
sin B
cos B
cosB $
sin B
cosB cot B ! cosB cot B
Glencoe/McGraw-Hill
180
Advanced Mathematical Concepts
Chapter 7
25. sin ! cos ! tan ! # cos2 !
!
1
sin ! cos ! $ # cos2 !
!
sin !
cos !
26.
1 " cos x
$$
1 # cos x
1
1
cos x
cos2 x
$
$
$
$
"
2
%
#
sin2 x
sin x
sin x
sin2 x
1 " cos x
! $$
1 # cos x
2
1 " 2 cos x # cos x
1 " cos x
$$$
! $$
2
sin x
1 # cos x
2
(1 " cos x)
1 " cos x
$$
! $$
2
1 " cos x
1 # cos x
2
(1 " cos x)
1 " cos x
$$$ ! $$
(1 " cos x)(1 # cos x)
1 # cos x
1 " cos x
1 " cos x
$$ ! $$
1 # cos x
1 # cos x
28.
sin ! # cos ! # tan ! sin !
! sec ! # cos ! tan !
cos x
sin x
$$ # $$
1 " tan x
1 " cot x
cos x
sin x
! $$
sin x # $$
cos x
1 " $$
1 " $$
sin !
cos !
!
sin ! # cos ! # $ sin !
! sec ! # cos ! tan !
sin2 !
cos !
sin x
sin ! # cos ! # $
cos x
cos x
$$
#
sin x % $
1 " $$ cos x
! sec ! # cos ! tan !
cos x
cos2
!
cos !
! sec ! # cos ! tan !
sin x
cos2
cos2 x
sin2 x
! $$ # $$
cos x " sin x
sin x " cos x
! sec ! # cos ! tan !
sin2
1
cos !
sin ! # $ ! sec ! # cos ! tan !
sin ! # sec ! ! sec ! # cos ! tan !
cos !
cos !
sin ! $ # sec !
! sec ! #
cos !sin ! "cos ! # sec !
! sec ! #
cos ! tan ! # sec !
! sec ! #
sec ! # cos ! tan !
! sec ! #
! sin x # cos x
Glencoe/McGraw-Hill
! # sin2 !
cos !
sin ! # $$
x
x
$$ # $$
sin x " cos x
sin x " cos x
2
2
sin x " cos x
! $$
sin x " cos x
(sin x # cos x)(sin x " cos x)
! $$$$
sin x " cos x
!
sin2 !
cos !
sin ! # $ # $
sin x
sin x
$$
cos x % $
1 " $$ sin x
cos2
1 " cos x
$$
1 # cos x
!
27.
sin x # cos x
!
!
csc2 x " 2csc x cot x # cot2 x
1
sin2 ! # cos2 ! ! 1
1!1
cos x
(csc x " cot x)2
181
cos ! tan !
cos ! tan !
cos ! tan !
cos ! tan !
Advanced Mathematical Concepts
Chapter 7
29. Sample answer: sec x !
!"2
30. Sample answer: tan x ! 2
1
2
31. Sample answer: cos x ! 0
32. Sample answer: sin x ! $
33.
35.
37.
39.
34. Sample answer: cot x ! 1
36. no
38. yes
40a. P ! I 02 R(1 " cos2 2&ft)
Sample answer: sin x ! 1
1
yes
no
I 2R
csc 2&ft
0
40b. P ! $$
2
1
2
41. f (x) ! $ sin !
42.
sin a
sin a ! sin % sin c ⇒ sin % ! $
sin c
cos $
cos b ! $
⇒ cos $ ! sin % cos b
sin %
cos c
cos a
cos c ! cos a cos b ⇒ cos b ! $
Then cos $ ! sin % cos b
sin a cos c
sin c cos a
cos c
sin a
! $ % $
cos a
sin c
! $%$
! tan a cot c
gx2
1
2
43. y ! " $2 (1 # tan2 !) # x tan !
44. Sample answer: $ tan ! sin2 !
45. By the Law of Sines,
46. 1
2v 0
a sin $
b
a
$ ! $ , so b ! $ .
sin %
sin $
sin %
1
Then A ! $ ab sin #
2
a sin $
1
A ! $a $
sin #
sin %
2
a 2 sin $ sin #
A ! $$
2 sin %
a 2 sin $ sin #
A ! $$$
2 sin(180° " ($ # #))
a 2 sin $ sin #
A ! $$
2sin($ # #)
#
Glencoe/McGraw-Hill
$
182
Advanced Mathematical Concepts
Chapter 7