082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

354 Chapter 9 Complex numbers
2 See FigureS.20.
y
y
y
3
Im(z) , 3
3
|z| . 3
y
0 arg(z) p/2
Re(z) 0
x
x
x
x
(a)
(b)
(c)
(d)
3
y
y
2
|z+j| . 3
y
|z –1| , |z –2|
–2
x
|z+2| 3
–1
3
x
1 2
x
(e)
(f)
(g)
FigureS.20
3 See FigureS.21.
v v v
2
s > 0
–2 < v < 2
s < 0
s s s
–2
(a) (b) (c)
FigureS.21
REVIEWEXERCISES9
1 Show that
1
cosθ−jsinθ =cosθ+jsinθ.
2 Expressin Cartesian form
5+4j
(a)
5−4j
1
(c)
2+3j + 1
2−3j
1
(b)
2+3j
1
(d)
x−jy
3 Find the modulusandargumentof −j, −3,1 +j,
cosθ+jsinθ.
4 Markonan Arganddiagramvectors corresponding to
the followingcomplex numbers: −3 +2j, −3 −2j,
cosπ+jsinπ.
5 Expressin the forma +bj:
(a) (cosθ +jsinθ) 6
(c)
cosθ+jsinθ
cosφ+jsinφ
6 Ifz ∈ C,showthat
(a) z+z =2Re(z)
(c) zz = |z| 2
(b)
1
(cosθ +jsinθ) 3
(b) z −z =2jIm(z)
7 Show thate jωt +e −jωt = 2cosωt andfindan
expression fore jωt −e −jωt .
8 Express1 +e 2jωt in the forma +bj.