PHS_Algebra_2_Chapter_13 4/7/06 9:34 AM Page 339
Answers for Lesson 13-8
1. 1.02
2. 1.02
3. -0.70
4. -1.06
5. 34
18
6. 13
7. 253
8. 234
10. !3
3
9. !2
11. 0
12. -1
14. 2 !3
3
17. undefined
15. undefined
19. 2
20. !2
21. -7.02
22. -1
23. 1
24. -1
25. -1.25
26. 17.13
27. 1.73
13. undefined
16. !3
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Exercises
18. 2
28. 1.02
29.
30.
y
2
O
π
2π
O
2
31.
π
2π
2
32.
y
O
y
π
2π
y
1
O
2
π
2π
2
33. 1.1547
34. 5.7588
35. -2.9238
36. 2
37. 1.0642
38. 1.3054
39. 1.7321
40. 0.5774
Algebra 2
Chapter 13
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PHS_Algebra_2_Chapter_13 4/7/06 9:34 AM Page 340
Answers for Lesson 13-8
41. a.
Exercises (cont.)
y
60
40
20
O
90
180
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Note that the units on the horizontal axis are degrees.
b. <28.3 ft
c. <23.1 ft
d. <20.7 ft
43. 2 !3
3 ; 1.15
45. -1
42. -1
44. -1
46.
y
π
O
48.
2π
47.
y
8
2
4
4
O
49.
y
O
1
3
2π
5 2
y
O
π
51. C
2π
1
2
50. B
π
52. A
2π
53. D
54. a. domain: all real numbers except multiples of p; range: all
real numbers $1 or #-1; period 2p
b. 1
c. -1
55. a. Reciprocals have the same sign.
b. The reciprocal of -1 is -1.
Algebra 2
Chapter 13
340
PHS_Algebra_2_Chapter_13 4/7/06 5:14 PM Page 341
Answers for Lesson 13-8 Exercises (cont.)
56. csc 1808 is undefined because sin 1808 is 0 and csc u = sin1 u.
57. sec 908 is undefined because cos 908 is 0 and sec u = cos1 u.
58. cot 08 is undefined because tan 08 is 0 and cot u = tan1 u.
59. a.
120
y
80
40
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O
45°
b. 63.9 ft
c. 69.3 ft
d. <41.48; 60.9 ft
60. a.
2
y
π
2π
x
b. The domain of y = tan x is all real numbers except odd
multiples of p2 , which are its asymptotes. The domain of
y = cot x is all real numbers except multiples of p, which
are its asymptotes. The range of both functions is all real
numbers.
c. The graphs have the same period and range. Their
asymptotes are shifted by p2 .
d. Answers may vary. Sample: x = p4 , x = 3p
4
Algebra 2
Chapter 13
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PHS_Algebra_2_Chapter_13 4/7/06 9:34 AM Page 342
Answers for Lesson 13-8
61.
Exercises (cont.)
62.
y
y
2
2
O
π
O
2
2
p
2
3 units up
63.
x
π
x
2π
64.
y
units left
2
y
x
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x
π
O
2
2
1 unit down
p
2
65.
y
4
2
67.
units right
66.
y
O
π
O
x
2π
π
O
π
x
2π
2
4
4 units right
2 units left, 1 unit down
68.
y
4
O
π
x
2π
x
p units left, 3 units up
Algebra 2
π
4
2
O
y
p
6
units right, 2 units down
Chapter 13
342
PHS_Algebra_2_Chapter_13 4/25/06 11:34 AM Page 343
Answers for Lesson 13-8 Exercises (cont.)
69. a.
y
1
O
π
x
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1
b. y = -cos x—domain: all real numbers; range: all real
numbers between -1 and 1, inclusive; period: 2p;
y = -sec x—domain: all real numbers except odd
multiples of p2 ; range: all real numbers except those
between -1 and 1; period: 2p
c. Multiples of p. By definition, sec x = cos1 x, so -cos x =
-sec x is equivalent to -cos x = 2cos1 x, or (cos x)2 = 1.
The solutions of (cos x)2 = 1 are the values of x for
which cos x = 1 or cos x = -1. These values are the
multiples of p.
d. Answers may vary. Sample: The graphs have the same
period and their signs are always the same. However,
they have no range values in common except 1 and -1.
e. The signs of -sec x and -cos x are the same because
reciprocals have the same sign.
70. a. II
b. I
71. y = sec x and y = csc x are not parabolas because parabolas
are not restricted by asymptotes, whereas the branches of
y = sec x and y = csc x are between asymptotes.
72. y = cos 3x has 3 cycles for each cycle of y = cos x. Thus, for
each cycle of y = sec x, y = sec 3x has 3 cycles. Each cycle
of y = sec 3x is 13 as wide as one cycle of y = sec x.
Algebra 2
Chapter 13
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PHS_Algebra_2_Chapter_13 4/18/06 11:52 AM Page 344
Answers for Lesson 13-8 Exercises (cont.)
73. a.
y
2
π
O
x
ⴚ2
b. Answers may vary. Sample: Given y = cot bx, as ∆b∆
decreases, the period increases; as ∆b∆ increases, the
period decreases. If b , 0, y = cot bx begins each branch
with negative y-values and ends with positive y-values;
the opposite is true for b . 0.
74. a.
y
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2
π
O
x
2π
2
b. Answers may vary. Sample: As ∆b∆ . 1 increases, the
graph of y = b sec x stretches vertically away from the
x-axis. As ∆b∆ decreases from 1 to 0, the graph shrinks
vertically toward the x-axis. For b , 0, the graph of
y = b sec x is a reflection of y = ∆b∆ sec x across the
x-axis. The asymptotes remain the same for all values of
b.
Algebra 2
Chapter 13
344