Solutions to exercises 639
Section 7.8
Chapter 8
Section 8.2
- 61 − 12 i 2. 4 3. 6 i 4. 181 + 14 i
5.− 81 − 16 i 6. 411 − 138 i 7. 10
- (i) 31 + 12 i (ii) 13
(iii)
9.z 1 = 1 ± 11 + 12 i 10.
12.−i
Section 8.3
- (i) 2 (ii)π 22
(iii)2(cos 1 π 221 + 1 i 1 sin 1 π 2 2)
- (i) 3 (ii)π
(iii)3(cos 1 π 1 + 1 i 1 sin 1 π)
- (i) (ii)−π 24
(iii) (cos 1 π 241 − 1 i 1 sin 1 π 2 4)
- (i) 2 (ii)π 26
(iii)2(cos 1 π 261 + 1 i 1 sin 1 π 2 6)
- (i) (ii) 3 π 24
(iii) (cos 13 π 241 + 1 i 1 sin 13 π 2 4)
- (i) 4 (ii) 4 π 23
(iii)4(cos 14 π 231 + 1 i 1 sin 14 π 2 3)
- (i) 1 (ii) 3 π 22
(iii)cos 13 π 221 + 1 i 1 sin 13 π 22
- (i)
(ii)
(iii)
3
26 6
cos sin
ππ
−
i
2
36 6
cos sin
ππ
i
6
5
6
5
6
cos sin
ππ
i
62
62
2
2
12
25
24
25
- i
5
13
12
13
- i
5
34
3
34
− i
−±21 3,i
13 ±i
3
1
2
2
1
2
=+−= −(), ()zz
i
zz
A
xxx
+−+−
246
246!!!
1
246
246
−+−+
xxx
!!!
x
xxx
−+ −
4
3
16
5
64
7
357
!!!
- (i)
(ii)
(iii)
- (i) 81 i (ii)−i 281
- 1281 cos
8
1 x 1 − 12561 cos
6
1 x 1 + 11601 cos
4
1 x 1
− 1321 cos
2
1 x 1 + 11
Section 8.4
- (i)f(x) 1 = 1 (3x
2
1 + 1 x 1 − 1 2) 1 + 1 i(2x 1 + 1 2)
(ii)g(x) 1 = 10 whenx 1 = 1223 andx 1 = 1 − 1 ,
h(x) 1 = 10 whenx 1 = 1 − 1 ,f(x) 1 = 10 when
x 1 = 1 − 1
(iii)(x 1 + 1 1)
2
(9x
2
1 − 112 x 1 + 1 8)
- (i)f(z) 1 = 1 [x
2
1 − 1 y
2
1 − 12 x 1 + 1 3] 1 + 1 i[2y(x 1 − 1 1)]
(ii)h(x, y) 1 = 10 wheny 1 = 1 0 or x 1 = 11 ,
g(x, y) 1 = 10 when (real) x 1 = 11 ,
f(z) 1 = 10 when
Section 8.5
- (i) (ii)
(iii)
- (i) 2 e
iπ 26(ii) 2 e
−iπ 26(iii)(1 2 2)e
−iπ 26- (i) 2 e
iπ 22(ii) 2 e
−iπ 22(iii)(1 2 2)e
−iπ 22- (i) 3 e
iπ(ii) 3 e
−iπ(iii)(1 2 3)e
−iπ
- 37.i 38.−i 39.− 1
41.cos 1 a 1 cosh 1 b 1 − 1 i 1 sin 1 a 1 sinh 1 b
- (i)
(ii)
Section 8.6
45.± 1 ,±i
- 1 ,cos 12 π 251 ± 1 i 1 sin 12 π 25 ,
cos 14 π 251 ± 1 i 1 sin 14 π 25
33
2
1
3
2
- 3
+−+
i
33
2
1
3
2
− 3
++
i
±−()12i
3 +i
1
2
3
2
− i
3
2
3
2
- i
12
()e
iπ 42 e
iπ 42 e
−iπ 4zi=± 12
y=± 2
1
512 12
cos sin
ππ
−
i
5
12 12
cos sin
ππ
i
5
17
12
17
12
cos sin
ππ
i