Chapter 7
466
Section 7.6 Solutions -------------------------------------------------------------------------------- 1.
[ ] [ ]1 1sin 2 cos sin(2 ) sin(2 ) sin 3 sin2 2
x x x x x x x x= + + − = +
3.
[ ]
[ ] [ ]
15sin 4 sin 6 5 cos(4 6 ) cos(4 6 )2
5 5cos( 2 ) cos10 cos 2 cos102 2
x x x x x x
x x x x
= ⋅ − − +
= − − = −
5.
[ ]
[ ] [ ]
14cos( ) cos 2 4 cos( 2 ) cos( 2 )2
2 cos cos( 3 ) 2 cos cos3
x x x x x x
x x x x
− = ⋅ − + + − −
= + − = +
7.
[ ] [ ]
3 5 1 3 5 3 5sin sin cos cos2 2 2 2 2 2 2
1 1cos( ) cos 4 cos cos 42 2
x x x x x x
x x x x
⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= − − +⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦
= − − = −
9. 2 4 1 2 4 2 4cos cos cos cos3 3 2 3 3 3 3
1 2 1 2cos 2 cos cos 2 cos2 3 2 3
x x x x x x
xx x x
⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + + −⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦
⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞= + − = +⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦
11. [ ] [ ]3 32 23cos(0.4 )cos(1.5 ) cos(1.9 ) cos( 1.1 ) cos(1.9 ) cos(1.1 )x x x x x x− = − + − = − +
13.
( ) ( ) ( ) ( ) ( ) ( )4sin 3 cos 3 3 2 sin 2 3 sin 4 3 2 sin 2 3 sin 4 3x x x x x x⎡ ⎤ ⎡ ⎤− = + − = −⎣ ⎦ ⎣ ⎦
15. 5 3 5 3cos5 cos3 2cos cos 2cos 4 cos
2 2x x x xx x x x+ −⎛ ⎞ ⎛ ⎞+ = =⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
17. 3 3sin 3 sin 2sin cos 2sin cos 2
2 2x x x xx x x x− +⎛ ⎞ ⎛ ⎞− = =⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
Section 7.6
467
19. 5 5
5 3 32 2 2 2sin sin 2sin cos 2sin( ) cos 2sin cos2 2 2 2 2 2
x x x xx x x xx x
⎛ ⎞ ⎛ ⎞− +⎜ ⎟ ⎜ ⎟⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞− = = − = −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
21. 2 7 2 7
2 7 3 3 3 3cos cos 2cos cos3 3 2 2
3 5 3 52cos cos 2cos cos2 6 2 6
x x x xx x
x x x x
⎛ ⎞ ⎛ ⎞+ −⎜ ⎟ ⎜ ⎟⎛ ⎞ ⎛ ⎞+ = ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= − =⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
23.
( ) ( )
( ) ( ) ( ) ( )
0.4 0.6 0.4 0.6sin 0.4 sin 0.6 2sin cos2 2
2sin 0.5 cos 0.1 2sin 0.5 cos 0.1
x x x xx x
x x x x
+ −⎛ ⎞ ⎛ ⎞+ = ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
= − =
25.
( ) ( )
( ) ( ) ( ) ( )
5 3 5 5 3 5sin 5 sin 3 5 2sin cos2 2
2sin 5 cos 2 5 2sin 5 cos 2 5
x x x xx x
x x x x
⎛ ⎞ ⎛ ⎞− +− = ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
= − = −
27.
4 6 4 6cos cos 2cos cos4 6 2 2
5 52cos cos 2cos cos24 24 24 24
x x x xx x
x x x x
π π π ππ π
π π π π
⎛ ⎞ ⎛ ⎞− + − −⎜ ⎟ ⎜ ⎟⎛ ⎞ ⎛ ⎞− + = ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= − − =⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
29.
2cos3 cossin 3 sin
x xx x
−−
=+
3sin2
x x+⎛ ⎞⎜ ⎟⎝ ⎠
3sin2
2
x x−⎛ ⎞⎜ ⎟⎝ ⎠
3sin2
x x+⎛ ⎞⎜ ⎟⎝ ⎠
3tan tan23cos
2
x x xx x
−⎛ ⎞= − = −⎜ ⎟− ⎝ ⎠⎛ ⎞
⎜ ⎟⎝ ⎠
Chapter 7
468
31.
2cos cos3sin 3 sin
x xx x
−−
=−
3 3sin sin2 2
2
x x x x+ −⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ sin 2 sin( ) sin 2 sin3 3 sin cos 2sin cos
2 2
x x x xx x x x x x
− −= =
− +⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
sin xtan 2
cos 2x
x=
33.
2cos5 cos 2sin 5 sin 2
x xx x+
=−
5 2 5 2cos cos2 2
x x x x− +⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
2 5 2 5 2sin cos2 2
x x x x− +⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
5 2 3cot cot2 2
x x x−⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
35.
2sin sincos cos
A BA B+
=+
sin cos2 2
A B A B+ −⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
2 cos cos2 2
A B A B+ −⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
tan2
A B+⎛ ⎞= ⎜ ⎟⎝ ⎠
37.
2cos cossin sin
A BA B
−−
=+
sin2
A B+⎛ ⎞⎜ ⎟⎝ ⎠
sin2
2
A B−⎛ ⎞⎜ ⎟⎝ ⎠
sin2
A B+⎛ ⎞⎜ ⎟⎝ ⎠
tan2
cos2
A BA B
−⎛ ⎞= − ⎜ ⎟− ⎝ ⎠⎛ ⎞
⎜ ⎟⎝ ⎠
39.
2sin sinsin sin
A BA B+
=−
sin cos2 2
2
A B A B+ −⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
sin cos2 2
sin cos cos sin2 2 2 2
tan cot2 2
A B A B
A B A B A B A B
A B A B
+ −⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠= ⋅
− + + −⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
+ −⎛ ⎞ ⎛ ⎞= ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
41.
2cos( ) cos( )sin( ) sin( )
A B A BA B A B+ + −
=+ + −
cos cos2 2
A B A B A B A B+ + − + − +⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
2 sin cos2 2
A B A B A B A B+ + − + − +⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
cos cotsin
A AA
= =
Section 7.6
469
43. Description of G note: ( )cos 2 (392)tπ Description of B note: ( )cos 2 (494)tπ
Combining the two notes: ( ) ( )cos 784 cos 988t tπ π+ . Using the sum-to-product identity then yields:
( ) ( )
( ) ( )( ) ( )
784 988 784 988cos 784 cos 988 2cos cos2 2
2cos 886 cos 1022cos 886 cos 102 (since cosine is even)
t t t tt t
t tt t
π π π ππ π
π ππ π
+ −⎛ ⎞ ⎛ ⎞+ = ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
= −=
The beat frequency is 494 – 392 = 102 Hz.
The average frequency is 494 392 443 Hz2+
= .
45. The resulting signal is
6 6
6 6 6 6
6 6 6
2 2sin sin1.55 10 0.63 10
1 1 1 11.55 10 0.63 10 1.55 10 0.63 102sin 2 cos 2
2 2
10 10 10 11.55 0.63 1.55
2sin 2 cos 22
tc tc
tc tc
tc tc
π π
π π
π π
− −
− − − −
⎛ ⎞ ⎛ ⎞+⎜ ⎟ ⎜ ⎟× ×⎝ ⎠ ⎝ ⎠⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞+ −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟× × × ×⎝ ⎠ ⎝ ⎠⎜ ⎟ ⎜ ⎟=⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎛ ⎞⎛ ⎞
+ −⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟=
⎜ ⎟⎜ ⎟⎝ ⎠
6
6 6
00.63
2
2 1 1 2 1 12sin 10 cos 102 1.55 0.63 2 1.55 0.63tc tcπ π
⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟
⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞= + −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠
47.
( ) ( )
( ) ( ) ( ) ( )
1540 2418 1540 2418sin 2 (770) sin 2 (1209) 2sin cos2 2
2sin 1979 cos 439 2sin 1979 cos 439
t t t tt t
t t t t
π π π ππ π
π π π π
+ −⎛ ⎞ ⎛ ⎞+ = ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
= − =
Chapter 7
470
49. Note that 52.5 7.5 180A+ + = , so that 120A = . So, the area is
( ) ( ) ( )( )
( ) ( )( )
( ) ( )( )
( ) ( )
( )
2
2
2
2 2 2
2 2
1100 cos 52.5 7.5 cos 52.5 7.510 ft. sin 52.5 sin 7.5 2 ft.2sin 120 2sin 120
50 cos 45 cos 60ft.
2sin 120
2 125 25 2 1 25 2 1 32 2ft. ft. ft.
33 32
25 6 3ft. 5.98 ft.
3
⎡ ⎤⋅ − − +⎣ ⎦=
⎡ ⎤−⎣ ⎦=
⎡ ⎤−⎢ ⎥ − −⎣ ⎦= = =
−= ≈
51. In the final step of the computation, note that cos cos cosA B AB≠ and sin sin sinA B AB≠ . Should have used the product-to-sum identities. 53. False. From the product-to-sum identities, we have
[ ]1cos cos cos( ) cos( )2
A B A B A B= + + − ,
and the right-side is not, in general, expressible as the cosine of a product. 55. True. From the product-to-sum identities, we have
[ ]1cos cos cos( ) cos( )2
A B A B A B= + + − .
57. Observe that [ ]
( )( )
( )
( ) ( )( )
( ) ( )( )
[ ]
sin sin sin sin sin sin
1 cos cos( ) sin2
1 cos sin cos( )sin21 1 sin sin2 2
1 sin sin2
1 sin( ) sin( ) sin( ) sin( )4
A B C A B C
A B A B C
A B C A B C
C A B C A B
C A B C A B
A B C C A B A B C A B C
=
⎡ ⎤= − − +⎢ ⎥⎣ ⎦
= − − +⎡ ⎤⎣ ⎦
⎧ ⎡ ⎤= + − + − −⎨ ⎣ ⎦⎩⎫⎡ ⎤− + + + − + ⎬⎣ ⎦⎭
= − + + − + − + + − + −
At this point, depending on which terms you decide to apply the odd identity for sine, the answer can take on a different form.
Section 7.6
471
59. [ ] [ ]1 1
2 2cos cos sin sin cos( ) cos( ) cos( ) cos( )cos( )
A B A B A B A B A B A BA B
− = + + − − − + +
= +
61. Observe that ( ) ( ) ( )
( ) ( )6 6 6
3 7 52 6 6
1 3sin( )sin 1 3 cos cos
1 cos cos
y x x x x x x
x x
π π π
π π
π π π⎡ ⎤= − − = − + − −⎣ ⎦⎡ ⎤= − −⎣ ⎦
The graph is as follows:
63. Observe that
( ) ( )( ) ( )( ) ( ) ( ) ( )
2 53 6
2 5 2 512 3 6 3 6
3 31 12 2 6 2 2 6
cos cos
cos cos
cos cos cos cos
y x x
x x x x
x x x x
π π
π π π π
π π π π
= −
⎡ ⎤= − + + −⎣ ⎦⎡ ⎤ ⎡ ⎤= − + − = − +⎣ ⎦ ⎣ ⎦
The graph is as follows:
Chapter 7
472
65. Consider the graph of 4sin cos cos 2y x x x= , as seen below:
From the graph, it seems as though this function is equivalent to sin 4x . We prove this identity below:
( )sin 2
4sin cos cos 2 2 2sin cos cos 2 2sin 2 cos 2 sin(2 2 ) sin 4x
x x x x x x x x x x=
= = = ⋅ =
67. To the right are the graphs of the following functions:
[ ]
1
2
13 2
sin 4 sin 2 (solid)sin 6 (dashed)
cos 2 cos 6
y x xy xy x x
==
= −
Note that the graphs of 1y and 3y are the same.
Section 7.7
473
Section 7.7 Solutions --------------------------------------------------------------------------------
1. The equation 2arccos2
θ⎛ ⎞
=⎜ ⎟⎜ ⎟⎝ ⎠
is equivalent to 2cos2
θ = . Since the range of
arccosine is [ ]0,π , we conclude that 4πθ = .
3. The equation 3arcsin2
θ⎛ ⎞− =⎜ ⎟⎜ ⎟⎝ ⎠
is equivalent to 3sin2
θ = − . Since the range of arcsine
is ,2 2π π⎡ ⎤−⎢ ⎥⎣ ⎦
, we conclude 3πθ = − .
5. The equation ( )1cot 1 θ− − = is equivalent to 2 cos2cot 1
sin22
θθθ
= − = − = . Since the
range of inverse cotangent is ( )0,π , we conclude that 34πθ = .
7. The equation 2 3arcsec3
θ⎛ ⎞
=⎜ ⎟⎜ ⎟⎝ ⎠
is equivalent to 2 3sec3
θ = , which is further the same
as 3 3cos22 3
θ = = . Since the range of inverse secant is 0, ,2 2π π π⎡ ⎞ ⎛ ⎤∪⎟ ⎜⎢ ⎥⎣ ⎠ ⎝ ⎦
, we conclude
that6πθ = .
9. The equation ( )1csc 2 θ− = is equivalent to csc 2θ = , which is further the same as 1sin2
θ = . Since the range of inverse cosecant is ,0 0,2 2π π⎡ ⎞ ⎛ ⎤− ∪⎟ ⎜⎢ ⎥⎣ ⎠ ⎝ ⎦
, we conclude
that6πθ = .
11. The equation ( )arc tan 3 θ− = is equivalent to 3 sin2tan 3 1 cos2
θθθ
= − = − = . Since
the range of inverse tangent is ,2 2π π⎛ ⎞−⎜ ⎟
⎝ ⎠, we conclude that
3πθ = − .
13. The equation ( )arcsin 0 θ= is equivalent to sin 0θ = . Since the range of arcsine is
,2 2π π⎡ ⎤−⎢ ⎥⎣ ⎦
, we conclude 0θ = .
15. The equation ( )1sec 1 θ− − = is equivalent to sec 1θ = − , which is further the same as
cos 1θ = − . Since the range of inverse secant is 0, ,2 2π π π⎡ ⎞ ⎛ ⎤∪⎟ ⎜⎢ ⎥⎣ ⎠ ⎝ ⎦
, we conclude that θ π= .
Chapter 7
474
17. The equation 1 1cos2
θ− ⎛ ⎞ =⎜ ⎟⎝ ⎠
is equivalent to 1cos2
θ = . Since the range of arccosine is
[ ]0,π , we conclude that 3πθ = , which corresponds to 60θ = .
19. The equation 1 2sin2
θ− ⎛ ⎞=⎜ ⎟⎜ ⎟
⎝ ⎠ is equivalent to 2sin
2θ = . Since the range of arcsine
is ,2 2π π⎡ ⎤−⎢ ⎥⎣ ⎦
, we conclude that 4πθ = , which corresponds to 45θ = .
21. The equation 1 3cot3
θ− ⎛ ⎞− =⎜ ⎟⎜ ⎟⎝ ⎠
is equivalent to 13 cos2cot
3 sin32
θθθ
= − = − = . Since
the range of inverse cotangent is ( )0,π , we conclude that 23πθ = , which corresponds to
120θ = .
23. The equation 3arc tan3
θ⎛ ⎞
=⎜ ⎟⎜ ⎟⎝ ⎠
is equivalent to 13 sin2tan
3 cos32
θθθ
= = = . Since the
range of inverse tangent is ,2 2π π⎛ ⎞−⎜ ⎟
⎝ ⎠, we conclude that
6πθ = , which corresponds to
30θ = . 25. The equation ( )arccsc 2 θ− = is equivalent to csc 2θ = − , which is further the same as
1sin2
θ = − . Since the range of inverse cosecant is ,0 0,2 2π π⎡ ⎞ ⎛ ⎤− ∪⎟ ⎜⎢ ⎥⎣ ⎠ ⎝ ⎦
, we conclude
that6πθ = − , which corresponds to 30θ = − .
27. The equation ( )arcsec 2 θ− = is equivalent to sec 2θ = − , which is further the
same as 1cos2
θ = − . Since the range of inverse secant is 0, ,2 2π π π⎡ ⎞ ⎛ ⎤∪⎟ ⎜⎢ ⎥⎣ ⎠ ⎝ ⎦
, we conclude
that 34πθ = , which corresponds to 135θ = .
29. The equation ( )1sin 1 θ− − = is equivalent to sin 1θ = − . Since the range of arcsine is
,2 2π π⎡ ⎤−⎢ ⎥⎣ ⎦
, we conclude that 2πθ = − , which corresponds to 90θ = − .
Section 7.7
475
31. The equation ( )arccot 0 θ= is equivalent to coscot 0sin
θθθ
= = , which implies
cos 0θ = . Since the range of inverse cotangent is ( )0,π , we conclude that 2πθ = , which
corresponds to 90θ = .
33. ( )1cos 0.5432 57.10− ≈ 35. ( )1tan 1.895 62.18− ≈ 37.
( )1 1 1sec 1.4973 cos 48.101.4973
− − ⎛ ⎞= ≈⎜ ⎟⎝ ⎠
39.
( )1 1 1csc 3.7893 sin 15.303.7893
− − ⎛ ⎞− = ≈ −⎜ ⎟−⎝ ⎠
41.
( )1 1 1cot 4.2319 tan4.2319
180 13.30 166.70
π− − ⎛ ⎞− = + ⎜ ⎟−⎝ ⎠
≈ − =
43. ( )1sin 0.5878 0.63− − ≈ −
45. ( )1cos 0.1423 1.43− ≈ 47. ( )1tan 1.3242 0.92− ≈ 49.
( )1 1 1cot 0.5774 tan 2.090.5774
π− − ⎛ ⎞− = + ≈⎜ ⎟−⎝ ⎠
51.
( )1 1 1csc 3.2361 sin 0.313.2361
− − ⎛ ⎞= ≈⎜ ⎟⎝ ⎠
53. 1 5 5sin sin12 12π π− ⎛ ⎞⎛ ⎞ =⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
since 52 12 2π π π
− ≤ ≤ .
55. ( )( )1sin sin 1.03− is undefined since 1.03 is not in the domain of inverse sine.
57. Note that we need to use the angle θ in ,2 2π π⎡ ⎤−⎢ ⎥⎣ ⎦
such that 7sin sin6πθ ⎛ ⎞= −⎜ ⎟
⎝ ⎠ . To
this end, observe that 1 17sin sin sin sin6 6 6π π π− −⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞− = =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠
.
59. Note that we need to use the angle θ in [ ]0,π such that 4cos cos3πθ ⎛ ⎞= ⎜ ⎟
⎝ ⎠ . To this
end, observe that 1 14 2 2cos cos cos cos3 3 3π π π− −⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠
.
61. Since ( )1cot cot x x− = for all x−∞ < < ∞ , we see that ( )1cot cot 3 3− = .
63. Note that we need to use the angle θ in 0, ,2 2π π π⎡ ⎞ ⎛ ⎤∪⎟ ⎜⎢ ⎥⎣ ⎠ ⎝ ⎦
such that sec sec .3πθ ⎛ ⎞= −⎜ ⎟
⎝ ⎠
To this end, observe that 1 1sec sec sec sec3 3 3π π π− −⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞− = =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠
.
65. 1 1csc csc2
−⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
is undefined since 12
is not in the domain of inverse cosecant.
Chapter 7
476
67. Since ( )1cot cot x x− = for all x−∞ < < ∞ , we see that ( )1cot cot 0 0− = .
69. Since ( )1tan tan x x− = for all 2 2
xπ π− < < , we see that 1tan tan
4 4π π− ⎛ ⎞⎛ ⎞− = −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
.
71. Not possible 73. ( )( ) ( )1 18 21
3 33cot cot cotπ π− −= − =
75. ( ) ( )1 1 2154 2 4csc csc cscπ π− −= − = −
77. Let 1 3sin4
θ − ⎛ ⎞= ⎜ ⎟⎝ ⎠
. Then, 3sin4
θ = , as
shown in the diagram:
Using the Pythagorean Theorem, we see that 2 2 23 4z + = , so that 7z = .
Hence, 1 3 7cos sin cos4 4
θ−⎛ ⎞⎛ ⎞ = =⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠.
79. Let 1 12tan5
θ − ⎛ ⎞= ⎜ ⎟⎝ ⎠
. Then, 12tan5
θ = , as
shown in the diagram:
Using the Pythagorean Theorem, we see that
2 2 212 5 z+ = , so that 13z = .
Hence, 1 12 12sin tan sin5 13
θ−⎛ ⎞⎛ ⎞ = =⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠.
θ θ
Section 7.7
477
81. Let 1 3sin5
θ − ⎛ ⎞= ⎜ ⎟⎝ ⎠
. Then, 3sin5
θ = , as
shown in the diagram:
Using the Pythagorean Theorem, we see that 2 2 23 5z + = , so that 4z = .
Hence, 1 3 3tan sin tan5 4
θ−⎛ ⎞⎛ ⎞ = =⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠.
83. Let 1 2sin5
θ − ⎛ ⎞= ⎜ ⎟⎜ ⎟
⎝ ⎠. Then, 2sin
5θ = , as
shown in the diagram:
Using the Pythagorean Theorem, we see that
( )22 22 5z + = , so that 23z = . So,
1 3 1 5 5 23sec sin sec4 cos 2323
θθ
−⎛ ⎞⎛ ⎞ = = = =⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠.
85. Let 1 1cos4
θ − ⎛ ⎞= ⎜ ⎟⎝ ⎠
. Then, 1cos4
θ = , as
shown in the diagram:
Using the Pythagorean Theorem, we see that 2 2 21 4z + = , so that 15z = . Hence,
1 1 1 4 4 15csc cos csc4 sin 1515
θθ
−⎛ ⎞⎛ ⎞ = = = =⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
.
87. Let 1 60sin61
θ − ⎛ ⎞= ⎜ ⎟⎝ ⎠
. Then, 60sin61
θ = , as
shown in the diagram:
Using the Pythagorean Theorem, we see that
2 2 260 61z + = , so that 11z = . Hence,
1 60 11cot sin cot61 60
θ−⎛ ⎞⎛ ⎞ = =⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠.
θθ
2
θθ
Chapter 7
478
89. Use ( )sin 2i I f tπ= with f = 5 and I = 115. Find the smallest positive value of t for which i = 85. To this end, observe
( )
( )
1
115sin 2 5 8585sin 10
11585sin
115 0.02647610
t
t
t
π
π
π
−
⋅ =
=
⎛ ⎞⎜ ⎟⎝ ⎠= ≈
So, 0.026476 sec. 26 mst ≈ = . 91. Given that ( ) 12 2.4sin(0.017 1.377)H t t= + − , we must find the value of t for which
( ) 14.4H t = . To this end, we have
1
12 2.4sin(0.017 1.377) 14.4sin(0.017 1.377) 1
0.017 1.377 sin (1) 21.3772 173.4
0.017
tt
t
t
π
π
−
+ − =− =
− = =
+= ≈
Now, note that 173.4 151 22.4− = . As such, this corresponds to June 22-23. 93. We need to find the smallest value of t for which
( )12.5cos 0.157 2.5 0t + = , and the graph of the left-side is decreasing prior to this value. We solve this graphically. The solid graph corresponds to the left-side of the equation. We have:
Notice that the solution is approximately t = 11.3, or about 11 years.
Section 7.7
479
95. Consider the following diagram:
Let θ α β= + . Then,
( )
2 2
2
tan tantan tan1 tan tan
1 7 88
1 7 7 71
xx x xx x
x x x
α βθ α βα β+
= + =−
+= = =
− −⎛ ⎞⎛ ⎞− ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
97. Use the formula
12 tan1
2
kf dM
π
−⎡ ⎤⎛ ⎞⎜ ⎟⎢ ⎥⎝ ⎠⎢ ⎥= −
⎢ ⎥⎢ ⎥⎣ ⎦
with 2m 0.002 km and 4 kmf d= = = .
We have the following specific calculation:
k = 2 km: 1
1
22 tan0.002 0.001 141 2 tan 0.0007048 km 0.70m
2 2M π
π π
−
−
⎡ ⎤⎛ ⎞⎜ ⎟⎢ ⎥ ⎡ ⎤⎛ ⎞⎝ ⎠⎢ ⎥= − = − ≈ ≈⎜ ⎟⎢ ⎥⎝ ⎠⎢ ⎥ ⎣ ⎦
⎢ ⎥⎣ ⎦
k = 10 km: 1
1
102 tan0.002 0.001 541 2 tan 0.00024 km 0.24m
2 2M π
π π
−
−
⎡ ⎤⎛ ⎞⎜ ⎟⎢ ⎥ ⎡ ⎤⎛ ⎞⎝ ⎠⎢ ⎥= − = − ≈ ≈⎜ ⎟⎢ ⎥⎝ ⎠⎢ ⎥ ⎣ ⎦
⎢ ⎥⎣ ⎦
99. Consider the following diagram:
Since 300tan200 x
α =−
, we have
1 300tan200 x
α − ⎛ ⎞= ⎜ ⎟−⎝ ⎠.
Also, since 150tanx
β = , we have
1 150tanx
β − ⎛ ⎞= ⎜ ⎟⎝ ⎠
.
Therefore, since α β θ π+ + = , we see that
1 1300 150tan tan200 x x
θ π − −⎛ ⎞ ⎛ ⎞= − −⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎝ ⎠.
101. The identity ( )1sin sin x x− = is valid only for x in the interval ,2 2π π⎡ ⎤−⎢ ⎥⎣ ⎦
, not [ ]0,π .
α
β
θα β
Chapter 7
480
103. In general, 11
1cottan
xx
−−≠ .
105. False. Upon inspection of the graphs, the portion to the right of the y-axis, when reflected over the y-axis does not match up identically with the left portion, as seen below:
More precisely, note that for instance 1 1 1sec (1) cos 01
− − ⎛ ⎞= =⎜ ⎟⎝ ⎠
, while
1 1 1sec ( 1) cos1
π− − ⎛ ⎞− = − =⎜ ⎟⎝ ⎠
. As such, ( ) ( )1 1sec secx x− −≠ − , for all x in the domain of
inverse secant. 107. False. This holds only on a subset of the domain to which cosecant is restricted in order to define its inverse.
109. 1 1sec2
− ⎛ ⎞⎜ ⎟⎝ ⎠
does not exist since 12
is not in the domain of the inverse secant function
(which coincides with the range of the secant function).
111. In order to compute 1 12 1sin cos sin2 2
− −⎡ ⎤⎛ ⎞ ⎛ ⎞+ −⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎢ ⎥⎝ ⎠⎣ ⎦
, we first simplify both 1 2cos2
− ⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
and 1 1sin2
− ⎛ ⎞−⎜ ⎟⎝ ⎠
:
1 2 2If cos , then cos2 2
θ θ− ⎛ ⎞= =⎜ ⎟⎜ ⎟
⎝ ⎠. So,
4πθ = .
1 1 1If sin , then sin2 2
β β− ⎛ ⎞− = = −⎜ ⎟⎝ ⎠
. So, 6πβ = − .
Hence,
1 12 1sin cos sin sin sin cos sin cos2 2 4 6 4 6 6 4
2 3 1 2 6 22 2 2 2 4
π π π π π π− −⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ − = − = −⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎜ ⎟ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎢ ⎥⎝ ⎠⎣ ⎦
⎛ ⎞⎛ ⎞ ⎛ ⎞ −⎛ ⎞= − =⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠⎝ ⎠ ⎝ ⎠
113. In order to compute ( )1sin 2sin (1)− , first observe that
1If =sin (1), then sin 1.θ θ− = So, 2πθ = .
Hence, ( )1sin 2sin (1) sin 2 sin 02π π− ⎛ ⎞= ⋅ = =⎜ ⎟
⎝ ⎠.
Section 7.7
481
115. Consider the function ( ) 3 cos4
f x x π⎛ ⎞= + −⎜ ⎟⎝ ⎠
.
a. Note that this function has a phase shift of 4π units to the right. So, we take the
interval used to define 1cos x− , namely [ ]0,π and add 4π to both endpoints to get
the interval 5,4 4π π⎡ ⎤⎢ ⎥⎣ ⎦
. Note that f is, in fact, one-to-one on this interval.
b. Now, we determine a formula for 1f − , along with its domain:
( )
( )
1
1
3 cos4
3 cos4
cos 34
cos 34
y x
y x
y x
y x
π
π
π
π
−
−
⎛ ⎞= + −⎜ ⎟⎝ ⎠
⎛ ⎞− = −⎜ ⎟⎝ ⎠
− = −
+ − =
So, the equation of the inverse of f is given by: ( )1 1( ) cos 34
f x xπ− −= + −
The domain of 1f − is equal to the range of f. Since the amplitude of f is 1 and there is a vertical shift up of 3 units, we see that the range of f is [ ]2, 4 . Hence, the domain of 1f −
is [ ]2, 4 .
117. Consider the function ( )14 6( ) 2 cot 2f x x π= + − .
(a) We know that cot 2y x= is 1-1 on ( )20, π . As such, since ( )6cot 2y x π= − is simply a horizontal shift of coty x= to the right 12
π units, we conclude that it is 1-1 on the interval
( ) ( )512 2 12 12 120 , ,π π π π π− − = − . Since multiplying by a constant and shifting it vertically do not
affect whether or not it is 1-1, we conclude that f is 1-1 on this interval as well. (b) Restricting our attention to x values in ( )5
12 12,π π− , we determine the inverse as follows:
( )( )
( )( )
( )( )( )
14 6
14 6
14 6
6
16
16
1112 2
2 cot 2
2 cot 2
2 cot 2
4( 2) cot 2
cot 4( 2) 2
cot 4 8 2
cot 4 8
y x
x y
x y
x y
x y
x y
x y
π
π
π
π
π
π
π
−
−
−
= + −
= + −
− = −
− = −
− = −
+ − =
+ − =
Hence, the inverse is ( )1 1112 2( ) cot 4 8f x xπ− −= + − with domain .
Chapter 7
482
119. The graphs of the following two functions on the interval [ ]3,3− is below:
( )11 sin sinY x−= , 2Y x=
The graphs are different outside the interval [ ]1,1− because the identity ( )1sin sin x x− = only holds for 1 1x− ≤ ≤ . The results are different outside the interval [ ]1,1− because the identity ( )1cos cos x x− = only holds for 1 1x− ≤ ≤ .
121. The graphs of the following two functions on the interval ,2 2π π⎡ ⎤−⎢ ⎥⎣ ⎦
is below:
( )11 csc cscY x−= , 2Y x=
Observe that the graphs do indeed coincide on this interval. This occurs since
( )1csc csc x x− = holds when ,0 0,2 2
x π π⎡ ⎞ ⎛ ⎤∈ − ∪⎟ ⎜⎢ ⎥⎣ ⎠ ⎝ ⎦.
Section 7.8
483
123. From the given information, we have the following diagram:
a. ( )( )40 9 720
41 41 1681sin 2 2sin cos 2x x x= = − − =
b. ( )1 409tan 1.34948x −= ≈ . So,
( )sin 2 sin 2.69896 0.42832x = = c. Yes, the results in parts a. and b. are the same.
Section 7.8 Solutions --------------------------------------------------------------------------------
1. The values of θ in [ ]0, 2π that satisfy 2cos2
θ = − are θ = 3 5,4 4π π .
3. First, observe that csc 2θ = − is equivalent to 1sin2
θ = − . The values of θ in [ ]0,4π
that satisfy 1sin2
θ = − are
7 7 11 11 7 11 19 23, 2 , , 2 , , ,6 6 6 6 6 6 6 6π π π π π π π πθ π π= + + = .
5. The only way tan 0θ = is for sin 0θ = . The values of θ in that satisfysin 0θ = (and hence, the original equation) are θ = , where is an integern nπ .
7. The values of θ in [ ]0, 2π that satisfy 1sin 22
θ = − must satisfy
7 7 11 11 7 11 19 232 , 2 , , 2 , , ,6 6 6 6 6 6 6 6π π π π π π π πθ π π= + + = .
So, dividing all values by 2 yields the following values ofθ which satisfy the original equation:
7 11 19 23, , ,12 12 12 12π π π πθ =
Chapter 7
484
9. The values of θ in that satisfy 1sin2 2θ⎛ ⎞ = −⎜ ⎟⎝ ⎠
must satisfy
7 112 , 22 6 6
n nθ π ππ π= + + , where n is an integer.
So, multiplying all values by 2 yields the following values ofθ which satisfy the original equation:
7 11 7 112 2 , 2 2 4 , 46 6 3 3
7 114 , 4 , where is an integer .3 3
n n n n
n n n
π π π πθ π π π π
π ππ π
⎛ ⎞ ⎛ ⎞= + + = + +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
= + +
11. The values of θ in [ ]2 , 2π π− that satisfy tan 2 3θ = must satisfy 4 4 2 2 5 52 , 2 , , 2 , , 2 , , 2
3 3 3 3 3 3 3 34 7 10 2 5 8 11, , , , , , ,
3 3 3 3 3 3 3 3
π π π π π π π πθ π π π π
π π π π π π π π
= + + − − − − − −
= − − − −.
So, dividing all values by 2 yields the following values ofθ which satisfy the original equation:
4 7 10 2 5 8 11, , , , , , ,6 6 6 6 6 6 6 6
2 7 5 5 4 11, , , , , , ,6 3 6 3 3 6 3 6
π π π π π π π πθ
π π π π π π π π
= − − − −
= − − − −
13. First, observe that sec 2θ = − is equivalent to 1cos2
θ = − . The values of θ in
[ ]2 ,0π− that satisfy 1cos2
θ = − are 2 4,3 3π πθ = − − .
15. The values of θ in that satisfy123
2
3cot 43
θ = − = − must satisfy
2 54 2 , 23 3
n nπ πθ π π= + + , where n is an integer.
This reduces to 243
nπθ π= + , where n is an integer.
So, dividing all values by 4 yields the following values ofθ which satisfy the original equation:
( )2 3, where is an integer
6 4 12nn n
ππ πθ+
= + = .
Section 7.8
485
17. First, observe that sec3 1θ = − is equivalent to cos3 1θ = − . The values of θ in for which this is true must satisfy
3 2 (2 1)n nθ π π π= + = + , where n is an integer. So, dividing all values by 5 yields the following values ofθ which satisfy the original equation:
(2 1) , where is an integer3
n nπθ += .
19. First, observe that csc3 1θ = is equivalent to sin 3 1θ = . The values of θ in for which this is true must satisfy
(4 1)3 22 2
nnπ πθ π += + = , where n is an integer.
So, dividing all values by 5 yields the following values ofθ in [ ]2 ,0π− which satisfy the original equation:
7 11, ,2 6 6π π πθ = − − −
21. First, observe that 2sin 2 3θ = is equivalent to 3sin 22
θ = . The values of θ in
[ ]0, 2π that satisfy 3sin 22
θ = must satisfy
2 2 2 7 82 , 2 , , 2 , , ,3 3 3 3 3 3 3 3π π π π π π π πθ π π= + + = .
So, dividing all values by 2 yields the following values ofθ which satisfy the original equation:
2 7 8 7 4, , , , , ,6 6 6 6 6 3 6 3π π π π π π π πθ = = .
23. First, observe that 3 tan 2 3 0θ − = is equivalent to 3tan 23
θ = . The values of θ in
[ ]0, 2π that satisfy 3tan 23
θ = must satisfy
7 7 7 13 192 , 2 , , 2 , , ,6 6 6 6 6 6 6 6π π π π π π π πθ π π= + + = .
So, dividing all values by 2 yields the following values ofθ which satisfy the original equation:
7 13 19, , ,12 12 12 12π π π πθ = .
Chapter 7
486
25. First, observe that ( )2cos 2 1 0θ + = is equivalent to ( ) 1cos 22
θ = − . The values of θ
in [ ]0, 2π that satisfy ( ) 1cos 22
θ = − must satisfy
2 2 4 4 2 4 8 102 , 2 , , 2 , , ,3 3 3 3 3 3 3 3π π π π π π π πθ π π= + + = .
So, dividing all values by 2 yields the following values ofθ which satisfy the original equation:
2 4 8 10 2 4 5, , , , , ,6 6 6 6 3 3 3 3π π π π π π π πθ = =
27. First, observe that 3 cot 3 02θ⎛ ⎞ − =⎜ ⎟⎝ ⎠
is equivalent to
cos13 22cot2 3 3 sin2 2
θθ
θ
⎛ ⎞⎜ ⎟⎛ ⎞ ⎝ ⎠= = =⎜ ⎟ ⎛ ⎞⎝ ⎠⎜ ⎟⎝ ⎠
.
The value of θ in[ ]0,2π that satisfy this equation must satisfy 2 3θ π= , so that 2
3πθ = .
29. Factoring the left-side of 2tan 1 0θ − = yields the equivalent equation ( )( )tan 1 tan 1 0θ θ− + = which is satisfied when either tan 1 0 or tan 1 0θ θ− = + = . The
values of θ in [ ]0, 2π that satisfy tan 1θ = are 5,4 4π πθ = , and those which satisfy
tan 1θ = − are 3 7,4 4π πθ = . Thus, the solutions to the original equation are
3 5 7, , ,4 4 4 4π π π πθ = .
31. Factoring the left-side of 22cos cos 0θ θ− = yields the equivalent equation ( )cos 2cos 1 0θ θ − = which is satisfied when either cos 0 or 2cos 1 0θ θ= − = . The
values of θ in [ ]0, 2π that satisfy cos 0θ = are 3,2 2π πθ = , and those which satisfy
1cos2
θ = are 5,3 3π πθ = . Thus, the solutions to the original equation are
3 5, , ,2 2 3 3π π π πθ = .
Section 7.8
487
33. Factoring the left-side of 2csc 3csc 2 0θ θ+ + = yields the equivalent equation ( )( )csc 2 csc 1 0θ θ+ + = which is satisfied when either csc 2 0 or csc 1 0θ θ+ = + = . The
values of θ in [ ]0, 2π that satisfy csc 2θ = − (or equivalently 1sin2
θ = − ) are
7 11,6 6π πθ = , and those which satisfy csc 1θ = − (or equivalently sin 1θ = − ) are 3
2πθ =
. Thus, the solutions to the original equation are 7 11 3, ,6 6 2π π πθ = .
35. Factoring the left-side of 2sin 2sin 3 0θ θ+ − = yields the equivalent equation ( )( )sin 3 sin 1 0θ θ+ − = which is satisfied when either sin 3 0 or sin 1 0θ θ+ = − = . Note that the equationsin 3 0θ + = has no solution since –3 is not in the range of sine. The
value of θ in [ ]0, 2π that satisfies sin 1θ = is2πθ = .
37. Factoring the left-side of 2sec 1 0θ − = yields the equivalent equation ( )( )sec 1 sec 1 0θ θ− + = , which is satisfied whenever sec 1θ = ± , or equivalently
cos 1θ = ± . The values of θ in ( )0, 2π for which this is true are 0,θ π= .
39. Factoring the left-side of 2 43sec (2 ) 0θ − = yields the equivalent equation
2 2sec 2 sec 2 03 3
θ θ⎛ ⎞⎛ ⎞− + =⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
, which is satisfied whenever 2sec 23
θ = ± , or
equivalently 3cos 22
θ = ± . The values of θ for which this is true must satisfy
5 7 11 13 17 19 232 , , , , , , ,6 6 6 6 6 6 6 6π π π π π π π πθ =
So, dividing by 2 then yields the values of θ in [ ]0, 2π for which this is true:
5 7 11 13 17 19 23, , , , , , ,12 12 12 12 12 12 12 12π π π π π π π πθ =
Chapter 7
488
41. In order to find all of the values of θ in 0 ,360⎡ ⎤⎣ ⎦ that satisfy sin 2 0.7843θ = − , we proceed as follows: Step 1: Find the values of 2θ whose sine is 0.7843− . Indeed, observe that one solution is ( )1sin 0.7843 51.655− − ≈ − , which is in QIV. Since the angles we seek have positive measure, we use the representative 360 51.655 308.345− ≈ . A second solution occurs in QIII, and has value 180 51.655 231.655+ = . Step 2: Use periodicity to find all values of θ that satisfy the original equation. Using periodicity with the solutions obtained in Step 1, we see that
2 308.345 , 308.345 360 , 231.655 , 231.655 360308.345 , 668.345 , 231.655 , 591.655
θ = + +
=
and so, the solutions to the original equation are approximately: 115.83 , 295.83 , 154.17 , 334.17θ ≈
43. In order to find all of the values of θ in 0 ,360⎡ ⎤⎣ ⎦ that satisfy tan 0.23432θ⎛ ⎞ = −⎜ ⎟⎝ ⎠
,
we proceed as follows:
Step 1: Find the values of 2θ whose tangent is 0.2343− .
Indeed, observe that one solution is ( )1tan 0.2343 13.187− − ≈ − , which is in QIV. Since the angles we seek have positive measure, we use the representative 360 13.187 346.813− ≈ . A second solution occurs in QII, and has value 346.813 180 166.813− = . Step 2: Use Step 1 to find all values of θ that satisfy the original equation, and exclude any value of θ that satisfies the equation, but lies outside the interval 0 ,360⎡ ⎤⎣ ⎦ . The solutions obtained in Step 1 are
166.813 , 346.8132θ= .
When multiplied by 2, the solution corresponding to346.813 will no longer be in the interval. So, the solution to the original equation in 0 ,360⎡ ⎤⎣ ⎦ is approximately
333.63θ ≈ .
Section 7.8
489
45. In order to find all of the values of θ in 0 ,360⎡ ⎤⎣ ⎦ that satisfy 5cot 9 0θ − = , we proceed as follows:
Step 1: First, observe that the equation is equivalent to 9cot5
θ = .
Step 2: Find the values of θ whose cotangent is 95
.
Indeed, observe that one solution is 1 1 19 1 5cot tan tan 29.054695 95
− − −⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟= = ≈⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠
,
which is in QI. A second solution occurs in QIII, and has value 29.0546 180 209.0546+ = . Since the input of cotangent is simply θ , and not some multiple thereof, we conclude that the solutions to the original equation are approximately 29.05 , 209.05θ ≈ .
47. In order to find all of the values of θ in 0 ,360⎡ ⎤⎣ ⎦ that satisfy 4sin 2 0θ + = , we proceed as follows:
Step 1: First, observe that the equation is equivalent to 2sin4
θ = − .
Step 2: Find the values of θ whose sine is 24
− .
Indeed, observe that one solution is 1 2sin 20.70484
− ⎛ ⎞− ≈ −⎜ ⎟⎜ ⎟⎝ ⎠
, which is in QIV.
Since the angles we seek have positive measure, we use the representative 360 20.7048 339.30− ≈ . A second solution occurs in QIII, and has value 20.7048 180 200.70+ = . Since the input of sine is simply θ , and not some multiple thereof, we conclude that the solutions to the original equation are approximately 200.70 , 339.30θ ≈ .
Chapter 7
490
49. In order to find all of the values of θ in 0 ,360⎡ ⎤⎣ ⎦ that satisfy 24cos 5cos 6 0θ θ+ − = , we proceed as follows:
Step 1: Simplify the equation algebraically. Factoring the left-side of the equation yields:
( )( )
No solution
4cos 3 cos 2 04cos 3 0 or cos 2 0
3cos or cos 24
θ θθ θ
θ θ
− + =
− = + =
= = −
Step 2: Find the values of θ whose cosine is 34
.
Indeed, observe that one solution is 1 3cos 41.40964
− ⎛ ⎞ ≈⎜ ⎟⎝ ⎠
, which is in QI. A
second solution occurs in QIV, and has value 360 41.4096 318.59− = . Since the input of cosine is simply θ , and not some multiple thereof, we conclude that the solutions to the original equation are approximately 41.41 , 318.59θ ≈ .
Section 7.8
491
51. In order to find all of the values of θ in 0 ,360⎡ ⎤⎣ ⎦ that satisfy 26 tan tan 12 0θ θ− − = , we proceed as follows:
Step 1: Simplify the equation algebraically. Factoring the left-side of the equation yields:
( )( )3tan 4 2 tan 3 03tan 4 0 or 2 tan 3 0
4 3tan or tan3 2
θ θθ θ
θ θ
+ − =
+ = − =
= − =
Step 2: Solve 4tan3
θ = − .
To do so, we must find the values of θ whose tangent is 43
− .
Indeed, observe that one solution is 1 4tan 53.133
− ⎛ ⎞− ≈ −⎜ ⎟⎝ ⎠
, which is in QIV. Since
the angles we seek have positive measure, we use the representative 360 53.13 306.87− ≈ . A second solution occurs in QII, and has value 180 53.13 126.87− = . Since the input of tangent is simply θ , and not some multiple thereof, we conclude that the solutions to this equation are approximately 126.87 , 306.87θ ≈ .
Step 3: Solve 3tan2
θ =
To do so, we must find the values of θ whose tangent is 32
.
Indeed, observe that one solution is 1 3tan 56.312
− ⎛ ⎞ ≈⎜ ⎟⎝ ⎠
, which is in QI. A second
solution occurs in QIII, and has value 180 56.31 236.31+ = . Since the input of tangent is simply θ , and not some multiple thereof, we conclude that the solutions to this equation are approximately 56.31 , 236.31θ ≈ . Step 4: Conclude that the solutions to the original equation are
56.31 , 126.87 , 236.31 , 306.87θ ≈ .
Chapter 7
492
53. In order to find all of the values of θ in 0 ,360⎡ ⎤⎣ ⎦ that satisfy 215sin 2 sin 2 2 0θ θ+ − = , we proceed as follows:
Step 1: Simplify the equation algebraically. Factoring the left-side of the equation yields: ( )( )5sin 2 2 3sin 2 1 0
5sin 2 2 0 or 3sin 2 1 02 1sin 2 or sin 25 3
θ θθ θ
θ θ
+ − =
+ = − =
= − =
Step 2: Solve 2sin 25
θ = − .
Step a: Find the values of 2θ whose sine is 25
− .
Indeed, observe that one solution is 1 2sin 23.5785
− ⎛ ⎞− ≈ −⎜ ⎟⎝ ⎠
, which is in QIV. Since the
angles we seek have positive measure, we use the representative 360 23.578 336.42− ≈ . A second solution occurs in QIII, namely 180 23.578 203.578+ = .
Step b: Use periodicity to find all values of θ that satisfy 2sin 25
θ = − .
Using periodicity with the solutions obtained in Step a, we see that 2 203.578 , 203.578 360 , 336.42 , 336.42 360
203.578 , 563.578 , 336.42 , 696.42θ = + +
=
and so, the solutions are approximately: 101.79 , 281.79 , 168.21 , 348.21θ ≈
Step 3: Solve 1sin 23
θ = .
Step a: Find the values of 2θ whose sine is 13
.
Indeed, observe that one solution is 1 1sin 19.47123
− ⎛ ⎞ ≈⎜ ⎟⎝ ⎠
, which is in QI.
A second solution occurs in QII, and has value 180 19.4712 160.528− = .
Step b: Use periodicity to find all values of θ that satisfy 1sin 23
θ = .
Using periodicity with the solutions obtained in Step a, we see that 2 19.4712 , 19.4712 360 , 160.528 , 160.528 360
19.4712 , 160.528 , 379.4712 , 520.528θ = + +
=
and so, the solutions are approximately: 9.74 , 189.74 , 80.26 , 260.26θ ≈ Step 4: Conclude that the solutions to the original equation are
101.79 , 281.79 , 168.21 , 348.21 , 9.74 , 189.74 , 80.26 , 260.26θ ≈ .
Section 7.8
493
55. In order to find all of the values of θ in 0 ,360⎡ ⎤⎣ ⎦ that satisfy 2cos 6cos 1 0θ θ− + = , we proceed as follows: Step 1: Simplify the equation algebraically. Since the left-side does not factor nicely, we apply the quadratic formula (treating
cosθ as the variable): ( ) ( ) ( )( )( )
26 6 4 1 1 6 4 2cos 3 2 22 1 2
θ− − ± − − ±
= = = ±
So, θ is a solution to the original equation ifNo solution since 3 2 2 1
cos 3 2 2 or cos 3 2 2θ θ+ >
= + = − .
Step 2: Find the values of θ whose cosine is 3 2 2− . Indeed, observe that one solution is ( )1cos 3 2 2 80.1207− − ≈ , which is in QI. A
second solution occurs in QIV, and has value 360 80.1207 279.88− = . Since the input of cosine is simply θ , and not some multiple thereof, we conclude that the solutions to this equation are approximately 80.12 , 279.88θ ≈ .
Step 3: Conclude that the solutions to the original equation are 80.12 , 279.88θ ≈ .
Chapter 7
494
57. In order to find all of the values of θ in 0 ,360⎡ ⎤⎣ ⎦ that satisfy 22 tan tan 7 0θ θ− − = , we proceed as follows: Step 1: Simplify the equation algebraically. Since the left-side does not factor nicely, we apply the quadratic formula (treating cosθ as the variable):
( ) ( ) ( )( )( )
21 1 4 2 7 1 57tan2 2 4
θ− − ± − − − ±
= =
So, θ is a solution to the original equation if either 1 57 1 57tan or tan
4 4θ θ− += = .
Step 2: Solve 1 57tan4
θ −= .
To do so, we must find the values of θ whose tangent is 1 574
− .
Indeed, observe that one solution is 1 1 57tan 58.5874
− ⎛ ⎞−≈ −⎜ ⎟⎜ ⎟
⎝ ⎠, which is in QIV.
Since the angles we seek have positive measure, we use the representative 360 58.587 301.41− ≈ . A second solution occurs in QII, and has value 180 58.587 121.41− = . Since the input of tangent is simply θ , and not some multiple thereof, we conclude that the solutions to this equation are approximately 121.41 , 301.41θ ≈ .
Step 3: Solve 1 57tan4
θ +=
To do so, we must find the values of θ whose tangent is 1 574
+ .
Indeed, observe that one solution is 1 1 57tan 64.934
− ⎛ ⎞+≈⎜ ⎟⎜ ⎟
⎝ ⎠, which is in QI. A
second solution occurs in QIII, and has value 180 64.93 244.93+ = . Since the input of tangent is simply θ , and not some multiple thereof, we conclude that the solutions to this equation are approximately 64.93 , 244.93θ ≈ . Step 4: Conclude that the solutions to the original equation are
64.93 , 121.41 , 244.93 , 301.41θ ≈ .
Section 7.8
495
59. In order to find all of the values of θ in 0 ,360⎡ ⎤⎣ ⎦ that satisfy 2csc (3 ) 2 0θ − = , we proceed as follows: Step 1: Simplify the equation algebraically. Factoring the left-side of 2csc (3 ) 2 0θ − = yields the equivalent equation
( )( )csc3 2 csc3 2 0θ θ− + = , which is satisfied whenever csc3 2θ = ± , or
equivalently 2sin 32
θ = ± . The values of θ for which this is true must satisfy
3 5 73 2 , 2 , 2 , 24 4 4 4
n n n nπ π π πθ π π π π= + + + + , where n is an integer,
which reduces to
34 2
nπ πθ = + , where n is an integer.
So, dividing by 3 then yields the values of θ in [ ]0, 2π for which this is true. Step 2: Convert the values obtained in Step 1 to degrees.
15 , 45 , 75 , 105 , 135 , 165 , 195 , 225 ,255 , 285 ,315 , 345θ =
61. By inspection, the values of x in [ ]0, 2π that satisfy the equation sin cosx x= are
5,4 4
x π π= .
63. Observe that
2
2
2
sec cos 21 cos 2
cos1 cos 2cos
cos 2cos 1 0(cos 1) 0
cos 1 0cos 1
x x
xx
x xx x
xx
x
+ = −
+ = −
+ = −
+ + =
+ =+ =
= −
The value of x in [ ]0, 2π that satisfies the equation cos 1x = − is x π= . Substituting this value into the original equation shows that it is, in fact, a solution to the original equation.
Chapter 7
496
65. Observe that
( )
( )( )
( )
2
2
2
3sec tan3
1 sin 3cos cos 3
1 sin 3cos 3
1 sin 1cos 3
1 sin 1 sin 11 sin 3
1 sin
x x
xx x
xx
xx
x xx
x
− =
− =
−=
−=
− −=
−− ( )
( )1 sin
1 sin
x
x
−
− ( )131 sin
3(1 sin ) 1 sin4sin 2
1sin2
x
x xx
x
=+
− = +=
=
The values of x in [ ]0,2π that satisfy the
equation 1sin2
x = are 5,6 6
x π π= .
Substituting these values into the original
equation shows that while 6π is a solution,
56π is extraneous. Indeed, note that
15 5 1 2sec tan6 6 3 3
2 23 3
3 3
π π⎛ ⎞ ⎛ ⎞− = −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ − −
= − ≠
So, the only solution is 6π .
67. Observe that
( )
( )( )
( )
2
2
2
csc cot 31 cos 3
sin sin1 cos 3
sin1 cos
3sin
1 cos 1 cos3
1 cos1 cos
x xx
x xx
xxx
x xx
x
+ =
+ =
+=
+=
+ +=
−+ ( )
( )1 cos
1 cos
x
x
+
+ ( )( )
31 cos
3 1 cos 1 cos3 3cos 1 cos
4cos 21cos2
x
x xx xx
x
=−
− = +
− = +=
=
The values of x in [ ]0,2π that satisfy the
equation 1cos2
x = are 5,3 3
x π π= .
Substituting this value into the original
equation shows that while3π is a solution,
53π is extraneous. Indeed, note that
15 5 1 2csc cot3 3 3 3
2 23 3 33
π π⎛ ⎞ ⎛ ⎞+ = +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ − −
= − = − ≠
So, the only solution is 3π .
Section 7.8
497
69. Observe that
2
2
2
2sin csc 012sin 0
sin2sin 1 0
sin2sin 1 0
1sin2
x x
xx
xxx
x
− =
− =
−=
− =
=
1 1sin or sin2 2
x x= − =
The solutions to these equations in [ ]0, 2π
are x = 3 5 7, , ,4 4 4 4π π π π . Substituting
these into the original equation shows that they are all, in fact, solutions to the original equation.
71. Observe that
( )
sin 2 4cos2sin cos 4cos
2cos sin 2 0
x xx x x
x x
==
− =
No solution
cos 0 or sin 2 0cos 0 or sin 2
x xx x= − == =
The solutions to these equations in [ ]0, 2π
are x = 3,2 2π π . Substituting these into the
original equation shows that they are all, in fact, solutions to the original equation.
73. Observe that
( )
2 sin tansin2 sincos
2 sin cos sin
sin 2 cos 1 0
x xxxx
x x x
x x
=
=
=
− =
sin 0 or 2 cos 1 01sin 0 or cos2
x x
x x
= − =
= =
The solutions to these equations in [ ]0, 2π are x = 70, , 2 , ,4 4π ππ π . Substituting these
into the original equation shows that they are all, in fact, solutions to the original equation.
Chapter 7
498
75. Observe that
( ) ( )( )
tan 2 cotsin 2 coscos 2 sin
cos cos 2 sin 2 sin 0
cos 2 0cos3 0
x xx xx x
x x x x
x xx
=
=
− =
+ =
=
Note that the solutions of cos3 0x = are 3 3 33 , 2 , 4 , , 2 , 4
2 2 2 2 2 2x π π π π π ππ π π π= + + + +
so that 5 3 7 11, , , , ,
6 6 2 2 6 6x π π π π π π= .
Substituting these into the original equation shows that they are all, in fact, solutions to the original equation. 77. Observe that
( )
3 sec 4sin
3 4sincos
3 4sin cos
3 2 2sin cos
3 2sin 2
3 sin 22
x x
xx
x x
x x
x
x
=
=
=
=
=
=
Next, the solutions of 3sin 22
x = are
2 22 , 2 , , 23 3 3 3
x π π π ππ π= + +
so that 7 4, , ,
6 3 6 3x π π π π= .
Substituting all of these into the original equation shows that they are all, in fact, solutions to the original equation.
79. Observe that
( )
( )
2
2 2 2
2 2 2
2
2
1sin cos 241sin cos sin41sin 1 sin sin413sin 14
1sin4
x x
x x x
x x x
x
x
− = −
− − = −
− − − = −
− = −
=
1 1sin or sin2 2
x x= − =
The solutions to these equations in [ ]0, 2π
are x = 5 7 11, , ,6 6 6 6π π π π . Substituting
these into the original equation shows that they are all solutions to original equation.
Section 7.8
499
81. Observe that
( )
( )( )
2
2
2
cos 2sin 2 0
1 sin 2sin 2 0
sin 2sin 3 0sin 1 sin 3 0
x x
x x
x xx x
+ + =
− + + =
− − =
+ − =
No solution
sin 1 0 or sin 3 0sin 1 or sin 3
x xx x+ = − == − =
The solution to these equations in [ ]0,2π
is x = 32π . Substituting this into the
original equation shows that it is, in fact, a solution to the original equation.
83. Observe that
( )
( )( )
2
2
2
2
2sin 3cos 0
2 1 cos 3cos 0
2 2cos 3cos 02cos 3cos 2 0
2cos 1 cos 2 0
x x
x x
x xx x
x x
+ =
− + =
− + =
− − =
+ − =
No solution
2cos 1 0 or cos 2 01cos or cos 22
x x
x x
+ = − =
= − =
The solutions to these equations in [ ]0, 2π
are x = 2 4,3 3π π . Substituting these into the
original equation shows that they are, in fact, solutions to the original equation.
85. Observe that
( )( )( )
( )( )
2 2
2 2
2
cos 2 cos 0
cos sin cos 0
cos 1 cos cos 0
2cos cos 1 02cos 1 cos 1 0
x x
x x x
x x x
x xx x
+ =
− + =
− − + =
+ − =
− + =
2cos 1 0 or cos 1 0
1cos or cos 12
x x
x x
− = + =
= = −
The solutions to these equations in [ ]0, 2π
are x = 5, ,3 3π π π . Substituting these into
the original equation shows that they are, in fact, solutions to the original equation.
87. Observe that 14
12
sec 2 sin 21 sin 2
4cos 21 4sin 2 cos 21 2sin 4
sin 4
x x
xx
x xx
x
=
=
===
The solutions to this equation must satisfy 5 13 17 25 29 37 414 , , , , , , ,
6 6 6 6 6 6 6 6x π π π π π π π π=
and so, the solutions are 5 13 17 25 29 37 41, , , , , , ,
24 24 24 24 24 24 24 24x π π π π π π π π=
Chapter 7
500
89. In order to find all of the values of x in 0 ,360⎡ ⎤⎣ ⎦ that satisfy 1cos(2 ) sin 02
x x+ = , we
proceed as follows: Step 1: Simplify the equation algebraically.
( )
2 2
2 2
2
2
1 1cos(2 ) sin 0 cos sin sin 02 2
11 sin sin sin 02
4sin sin 2 0
( 1) ( 1) 4(4)( 2) 1 33sin2(4) 8
x x x x x
x x x
x x
x
+ = ⇒ − + =
⇒ − − + =
⇒ − − =
− − ± − − − ±⇒ = =
Step 2: Solve 1 33sin8
x += .
To do so, we must find the values of x whose sine is 1 338
+ .
Indeed, observe that one solution is 1 1 33sin 57.478
− ⎛ ⎞+≈⎜ ⎟⎜ ⎟
⎝ ⎠, which is in QI. A
second solution occurs in QII, and has value 180 57.47 122.53− = . Since the input of sine is simply x, and not some multiple thereof, we conclude that the solutions to this equation are approximately 57.47 , 122.53x ≈ .
Step 3: Solve 1 33sin8
x −= .
To do so, we must find the values of x whose sine is 1 338
− .
Indeed, observe that one solution is 1 1 33sin 36.388
− ⎛ ⎞−≈ −⎜ ⎟⎜ ⎟
⎝ ⎠, which is in QIV.
Since the angles we seek have positive measure, we use the representative 360 36.38 323.62− = . A second solution occurs in QIII, and has value 180 36.38 216.38+ = . Since the input of sine is simply x, and not some multiple thereof, we conclude that the solutions to this equation are approximately 216.38 , 323.62x ≈ . Step 4: Conclude that the solutions to the original equation are
57.47 , 122.53 , 216.38 , 323.62x ≈ . Substituting these into the original equation shows that they are, in fact, solutions to the original equation.
Section 7.8
501
91. In order to find all of the values of x in 0 ,360⎡ ⎤⎣ ⎦ that satisfy 26cos sin 5x x+ = , we proceed as follows: Step 1: Simplify the equation algebraically.
( )
( )( )
2
2
2
6cos sin 5
6 1 sin sin 5
6 6sin sin 53sin 1 2sin 1 0
x x
x x
x xx x
+ =
− + =
− + =
+ − =
1 13sin 1 0 or 2sin 1 0 so that sin or sin3 2
x x x x+ = − = = − =
Step 2: Solve 1sin3
x = − .
To do so, we must find the values of x whose sine is 13
− .
Indeed, observe that one solution is 1 1sin 19.473
− ⎛ ⎞− ≈ −⎜ ⎟⎝ ⎠
, which is in QIV.
Since the angles we seek have positive measure, we use the representative 360 19.47 340.53− = . A second solution occurs in QIII, and has value 180 19.47 199.47+ = . Since the input of sine is simply x, and not some multiple thereof, we conclude that the solutions to this equation are 199.47 , 340.53x ≈ .
Step 3: Solve 1sin2
x = .
Indeed, observe that one solution is 1 1sin 302
− ⎛ ⎞ =⎜ ⎟⎝ ⎠
, which is in QI. A second
solution occurs in QII, and has value 180 30 150− = . Since the input of sine is simply x, and not some multiple thereof, we see that the solutions are 30 , 150x = . Step 4: Conclude that the solutions to the original equation are
30 , 150 , 199.47 , 340.53x ≈ . Substituting these into the original equation shows that they are all solutions.
Chapter 7
502
93. In order to find all of the values of x in 0 ,360⎡ ⎤⎣ ⎦ that satisfy 2cot 3csc 3 0x x− − = , we proceed as follows: Step 1: Simplify the equation algebraically.
( )
( )( )
2
2
2
cot 3csc 3 0
csc 1 3csc 3 0
csc 3csc 4 0csc 1 csc 4 0
x x
x x
x xx x
− − =
− − − =
− − =
+ − =
csc 1 0 or csc 4 0csc 1 or csc 4
1sin 1 or sin4
x xx x
x x
+ = − == − =
= − =
Step 2: Solve sin 1x = − . To do so, we must find the values of x whose sine is 1− . Indeed, the only solution is ( )1sin 1 270− − = , which is in QIII.
Step 3: Solve 1sin4
x = .
Indeed, observe that one solution is 1 1sin 14.484
− ⎛ ⎞ ≈⎜ ⎟⎝ ⎠
, which is in QI. A second
solution occurs in QII, and has value 180 14.48 165.52− = . Since the input of sine is simply x, and not some multiple thereof, we conclude that the solutions to this equation are 14.48 , 165.52x ≈ . Step 4: Conclude that the solutions to the original equation are
14.48 , 165.52 ,270x ≈ . Substituting these into the original equation shows that they are, in fact, solutions to the original equation.
Section 7.8
503
95. In order to find all of the values of x in 0 ,360⎡ ⎤⎣ ⎦ that satisfy 22sin 2cos 1 0x x+ − = , we proceed as follows: Step 1: Simplify the equation algebraically.
( )2
2
2
2
2
2sin 2cos 1 0
2 1 cos 2cos 1 0
2 2cos 2cos 1 02cos 2cos 1 0
( 2) ( 2) 4(2)( 1) 2 2 3 1 3cos2(2) 4 2
x x
x x
x xx x
x
+ − =
− + − =
− + − =
− − =
− − ± − − − ± ±= = =
1 3No solution since 12
1 3 1 3cos or cos2 2
x x
+>
− += = .
Step 2: Solve 1 3cos2
x −= .
To do so, we must find the values of x whose cosine is 1 32− .
Indeed, the only solution is 1 1 3cos 111.472
− ⎛ ⎞−≈⎜ ⎟⎜ ⎟
⎝ ⎠, which is in QII. A second
solution occurs in QIII and has value 360 111.47 248.53− = . Since the input of cosine is simply x, and not some multiple thereof, we conclude that the solutions to this equation are approximately 111.47 , 248.53x ≈ . Step 3: Conclude that the solutions to the original equation are
111.47 , 248.53x ≈ . Substituting these into the original equation shows that they are, in fact, solutions to the original equation.
Chapter 7
504
97. Observe that ( ) ( )
( ) ( )( ) ( ) ( )( )( ) ( )
( ) ( )( )
( )
2 2116 4 4
1 14 4 4 4 4 4
14 4 4
14 4 4
1 14 2 4
12 2
csc cos 0
csc cos csc cos 0
csc cos
sin cos
sin 2
sin
x x
x x x x
x x
x x
x
x
− =
− + =
= ±
=±
=± ⋅
± =
The values of x that satisfy these equation must satisfy 5,2 6 6x π π= . So, the solutions are
5,3 3
x π π= .
99. Solving for x yields:
2,400 400sin 2,0006
400 400sin6
1 sin6
x
x
x
π
π
π
⎛ ⎞= +⎜ ⎟⎝ ⎠⎛ ⎞= ⎜ ⎟⎝ ⎠
⎛ ⎞= ⎜ ⎟⎝ ⎠
This equation is satisfied when 6 2
xπ π= , so that 6 3
2x π
π= ⋅ = . So, the sales reach 2,400
in March.
Section 7.8
505
101. Consider the following diagram:
Let h = height of the trapezoid, and x = length of one base and two edges of the trapezoid, as labeled above. Note that α β θ= = since they are alternate interior angles. As such,
sin hx
θ = , so that sinh x θ= .
Furthermore, using the Pythagorean Theorem enables us to find z: 2 2 2z h x+ = , so that ( )
2
2 2 2 2 2 2
cos
( sin ) 1 sin cosz x h x x x xθ
θ θ θ
=
= − = − = − = .
Since 0 θ π≤ ≤ , we conclude that cosz x θ= . Hence, 1b x= and ( )2 2 2 cos 1 2cosb x z x x xθ θ= + = + = + .
Thus, the area A of the cross-section of the rain gutter is
( ) ( ) ( )
( )[ ][ ]
( )
1 2
2 (1 cos )
2
2
2
1 1 sin 1 2cos2 2
sin (1 cos )
sin sin cos
1sin 2sin cos2sin 2sin
2
x
A h b b x x x
x x
x
x
x
θ
θ θ
θ θ
θ θ θ
θ θ θ
θθ
= +
= + = + +⎡ ⎤⎣ ⎦
= +
= +
⎡ ⎤= +⎢ ⎥⎣ ⎦⎡ ⎤= +⎢ ⎥⎣ ⎦
θθ
α β
Chapter 7
506
103. Solving the equation 200 100sin 3002
xπ⎛ ⎞+ =⎜ ⎟⎝ ⎠
, for 2,000x > yields:
200 100sin 3002
100sin 1002
sin 12
x
x
x
π
π
π
⎛ ⎞+ =⎜ ⎟⎝ ⎠⎛ ⎞ =⎜ ⎟⎝ ⎠⎛ ⎞ =⎜ ⎟⎝ ⎠
Observe that this equation is satisfied when 22 2
x nπ π π= + , where n is an integer, so that
21 2 1 4x n nππ⎛ ⎞= + = +⎜ ⎟⎝ ⎠
, where n is an integer. So, the first value of n for which
1 4 2,000n+ > is 500n = . The resulting year is 1 4(500) 2001x = + = . 105. Use ( ) ( )sin sini i r rn nθ θ= with the given information to obtain
( ) ( )1.00sin 75 2.417sin rθ= so that
( ) ( )
( )1
1.00sin 75sin
2.4171.00sin 75
24 sin2.417
r
r
θ
θ−
=
⎛ ⎞⎜ ⎟≈ =⎜ ⎟⎝ ⎠
107. Observe that using the identity
sin 2 2sin cosA A A= with 3
A xπ= yields
2sin cos 3 43 3
sin 2 13
x x
x
π π
π
⎛ ⎞ ⎛ ⎞ + =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
⎛ ⎞⋅ =⎜ ⎟⎝ ⎠
This equation is satisfied when 23 2
xπ π= , so
that 34
x = . So, it takes 3 sec.4
for the
volume of air to equal 4 liters.
Section 7.8
507
109. First, we solve 2sin 2sin 2 0x x− + = on [ ]0, 2π :
[ ]( )
2sin 2sin 2 02sin 2 2sin cos 0
2sin 1 2cos 0
x xx x x
x x
− + =
− + =
− + =
so that 2sin 0 or 1 2cos 0
12sin 0 or cos2
x x
x x
= − + =
= =
These equations are satisfied when 50, , 2 , ,3 3
x π ππ π= . We now need to determine the
corresponding y-coordinates. x ( ) 2cos cos 2y x x x= − Point 0 ( ) ( ) ( )0 2cos 0 cos 2 0 2 1 1y = − ⋅ = − = ( )0,1
3π ( ) ( ) ( ) 1 1 32cos cos 2 23 3 3 2 2 2
y π π π ⎛ ⎞ ⎛ ⎞= − ⋅ = − − =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
3,3 2π⎛ ⎞⎜ ⎟⎝ ⎠
53
π ( ) ( ) ( ) 1 1 35 5 52cos cos 2 23 3 3 2 2 2y π π π ⎛ ⎞ ⎛ ⎞= − ⋅ = − − =⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠ 5 3,
3 2π⎛ ⎞
⎜ ⎟⎝ ⎠
π ( ) ( ) ( )2cos cos 2 2( 1) 1 3y π π π= − ⋅ = − − = − ( ), 3π − 2π ( ) ( ) ( )2 2cos 2 cos 2 2 2(1) 1 1y π π π= − ⋅ = − = ( )2 ,1π
So, the turning points are ( )0,1 , 3,3 2π⎛ ⎞⎜ ⎟⎝ ⎠
, 5 3,3 2π⎛ ⎞
⎜ ⎟⎝ ⎠
, ( ), 3π − , and ( )2 ,1π .
111. The value 32
πθ = does not satisfy the original equation. Indeed, observe that
32 sin 2 1 12π⎛ ⎞+ = − =⎜ ⎟
⎝ ⎠, while 3sin 1
2π⎛ ⎞ = −⎜ ⎟
⎝ ⎠. So, this value of θ is an extraneous
solution. 113. Cannot divide by cos x since it could be zero. Rather, should factor as follows:
( )
6sin cos 2cos6sin cos 2cos 0
2cos 3sin 1 0
x x xx x x
x x
=− =
− =
Now, proceed…
115. False. For instance, 3sin2
θ = has two solutions on [ ]0,2π , namely 2,3 3π πθ = .
117. True. This follows by definition of an identity.
Chapter 7
508
119. Solving the equation 4 216sin 8sin 1 0θ θ− + = on [ ]0, 2π yields
( )( )( )
4 2
22
16sin 8sin 1 0
4sin 1 0
2sin 1 2sin 1 0
θ θ
θ
θ θ
− + =
− =
− + =
So, θ satisfies the original equation if either 2sin 1 0 or 2sin 1 0θ θ− = + = .
Observe that 2sin 1 0θ − = is equivalent to 1sin2
θ = , which is satisfied when 5,6 6π πθ = .
Also, 2sin 1 0θ + = is equivalent to 1sin2
θ = − , which is satisfied when 7 11,6 6π πθ = .
So, we conclude that the solutions to the original equation are 5 7 11, , ,6 6 6 6π π π πθ = .
121. First, observe that using the addition formulae for sin( )A B± yields
sin sin sin cos cos sin4 4 4 4
x x x xπ π π π⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ + − = +⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
sin cos cos sin4 4
x xπ π⎡ ⎤ ⎛ ⎞ ⎛ ⎞+ −⎢ ⎥ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦
( )2 2sin22 sin
x
x
⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦
=
=
We substitute this for the left-side of the given equation to obtain: 2sin sin
4 4 2
22 sin2
1sin2
x x
x
x
π π⎛ ⎞ ⎛ ⎞+ + − =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
=
=
The smallest positive value of x for which this is true is or 306
x π= .
123. Observe that using a half-angle formula, we see that ( )( ) ( )3 2
63
1 costan
1 cos
xx
x
−=
+, thereby
resulting in the equivalent equation ( )26tan 1x = − , which has no solution.
Section 7.8
509
125. Factoring the left-side of ( )44csc 4 0π θ π− − = yields the equivalent equation
( )( ) ( )( )2 24 4csc 2 csc 2 0π πθ π θ π− − − + = , which is satisfied whenever either
( )24csc 2 0π θ π− − = or ( )2
4csc 2 0π θ π− + = . The second equation has no solution since the left-side is always greater than or equal to 2. The first equation is equivalent to
( )2 14 2sin π θ π− = , which holds whenever ( ) 2
4 2sin π θ π− = ± . The values of θ for which this is true must satisfy
4 4 2nπ π πθ π− = + , where n is an integer.
Solving this equation for θ yields the values of θ in for which this is true: 5 2 , where is an integer .n nθ = +
127. Consider the graphs below of 1 2sin , cos 2y yθ θ= = .
Observe that the solutions of the equation sin cos 2θ θ= on [ ]0,π are approximately
( ) ( )1 0.524,0.5 , 2 2.618,0.5P P .
The exact solutions are 5,6 6π π .
129. Consider the graphs below of 1 2sin , secy yθ θ= = .
Since the curves never intersect, there are no solutions of the equation sin secθ θ= on [ ]0,π .
Chapter 7
510
131. Consider the graphs below of 1 2sin ,y y eθθ= = .
First, note that while there are no positive solutions of the equation sin eθθ = , there are infinitely many negative solutions (at least one between each consecutive pair of x-intercepts). They are all irrational, and there is no apparent closed-form formula that can be used to generate them.
133. To determine the smallest positive solution (approximately) of the equation sec3 csc 2 5x x+ = graphically, we search for the intersection points of the graphs of the following two functions:
1 2sec3 csc 2 , 5y x x y= + =
The x-coordinate of the intersection point is in radians. Observe that the smallest positive solution, in degrees, is approximately 7.39 .