Chapter 8

2(sec 2 x 2 tan 2 x)
9.
5 sin 2x sec x
csc x
2(1)
5 sin 2x sec x
csc x
2
5 sin 2x sec x
csc x
2 sin x 5 sin 2x sec x
2 sin x cos x
5 sin 2x sec x
cos x
sin 2x
5 sin 2x sec x
cos x
sin 2x sec x 5 sin 2x sec x
11p
10. a) x 5 7p or
6 6
5p
b) x 5 p or
4 4
4p
c) x 5 2p or
3 3
11. a) y 522 or 2
5p 7p 11p
b) x 5 p or
6 , 6 , 6 , 6
7p 11p
12. a) x 5 p or
2 , 6 , 6
p 5p 7p 11p
b) x 5 0, p, or 2p
6 , 6 , 6 , 6 ,
2p 4p 7p
c) x 5 p or
4 , 3 , 3 , 4
d) x 5 0.95 or 4.09
3p
13. x 5 p p, or
2 , 2
Chapter Self-Test, p. 441
1 2 2 sin 2 x
1.
1 sin x 5 cos x
cos x 1 sin x
1 2 2 sin 2 x
1 sin x 2 sin x
cos x 1 sin x
5 cos x 2 sin x
1 2 2 sin 2 x
5 cos x 2 sin x
cos x 1 sin x
1 2 2 sin 2 x 5 (cos x 2 sin x)
3(cos x 1 sin x)
cos 2x 5 (cos x 2 sin x)
3(cos x 1 sin x)
cos 2x 5 cos 2 x 2 sin 2 x
cos 2x 5 cos 2x
2. all real numbers x, where 0 # x # 2p
3. a) x 5 p or x 5 11p
6 6
b) x 5 2p or x 5 5p 3 3
c) x 5 5p or x 5 7p 4 4
4. a 5 2, b 5 1
5. t 5 7, 11, 19, and 23
11p
6. Nina can find the cosine of by using
4
the formula
cos (x 1 y) 5 cos x cos y 2 sin x sin y.
The cosine of p is 21, and the
7p !2
cosine of is Also, the sine of p is 0,
4 2 .
7p
and the sine of is 2 !2 Therefore,
4 2 .
cos 11p
4 5 cos ap 17p 4 b
5 a21 3 !2
!2
b 2 a0 3 2
2 2 b
52 !2
2 2 0
52 !2
2
7. x 5 3.31 or 6.12
8.
9.
2 33 2 16
65 , 65
a) 2 4!5
9
c)
1
22
b) d)
9
27
10.
p 5p
a) or
3 , 3 , 3 , 3
2p
b) or
3 , 3 , 3 ,
c) x 52pand
p
Chapter 8
0
Getting Started, p. 446
1. a)
1
5 5 1
2 25
d) ! 3 125 5 5
b) 1 e) 2 !121 5211
c) !36 5 6
2
3 27
f) a Å 8 b 5 9 4
2. a) 3 7 5 2187 d) 7 4 5 2401
b) (22) 2 5 4 e) 8 2 3 5 4
c) 10 3 5 1000 f) 4 1 2 5 !4 5 2
3. a) 8m 3 d) x 3 y
1
b)
a 8 b 10 e) 2d 2 c 2
3
c) 40 x f) x
4. a)
y
40
30
20
10
x
–2 –1 0 1 2 3 4
–10
D 5 5xPR6, R 5 5 yPR 0 y . 06,
y-intercept 1, horizontal
asymptote y 5 0
y
b)
40
30
20
10
x
–4 –3 –2 –1 0 1 2
–10
3 2 !5
Å 6
4p
3
D 5 5xPR6, R 5 5 yPR0 y . 06,
y-intercept 1, horizontal
asymptote y 5 0
c)
D 5 5xPR6, R 5 5 yPR y .226,
y-intercept 21, horizontal
asymptote y 522
5. a) i) y 5 x 1 6
3
ii) y 56!x 1 5
3 x
iii) y 5 Å 6
iv)
b) The inverses of (i) and (iii) are functions.
6. a) 800 bacteria
b) 6400 bacteria
c) 209 715 200
d) 4.4 3 10 15
7. 12 515 people
8.
Similarities
• same y-intercept
• same shape
• same horizontal
asymptote
• both are always
positive
Lesson 8.1, p. 451
1. a) x 5 4 y or f 21 (x) 5 log 4 x
1
–1
0
1 2 3 4
–1
–2
–3
–4
b) x 5 8 y or f 21 (x) 5 log 8 x
1
–1
0
1 2 3 4
–1
–2
–3
–4
80
60
40
20
x
–3 –2 –1
0
1 2 3 4
–20
y
y
y
Differences
• one is always increasing,
the other is always
decreasing
• different end behaviour
• reflections of each other
across the y-axis
x
x
Answers
NEL Answers 667