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

334 Chapter 9 Complex numbers
y
y
b
r
u
a
(a, b)
x
7– 4
p
p –4 –
1
2
–1 (1, –1)
x
Figure9.3
Polar andCartesian formsofacomplex
number.
Figure9.4
Argand diagram depictingz = 1 −jin
Example 9.11.
Note that
r̸
(−θ) =r(cos(−θ) +jsin(−θ))
=r(cosθ −jsinθ)
=z
Ifz=a+bjthenz=a−bjandz =r̸
(−θ).
Example9.11 Depict the complex numberz = 1 − j on an Argand diagram and convert it into polar
form.
Solution Therealpartofzis1andtheimaginarypartis −1.Wethereforeplotapointinthex--y
plane withx = 1 andy = −1 asshown inFigure 9.4.
FromFigure9.4weseethatr = √ 1 2 + (−1) 2 = √ 2and θ = −45 ◦ or −π/4radians.
Thereforez = 1 −j = √ 2̸ (−π/4).
To express a complex number in polar form it is essential to draw an Argand diagram
and notsimplyquoteformulae, asthe following example will show.
Example9.12 Expressz = −1 −j inpolar form.
Solution If we use the formula |z| =r= √ a 2 +b 2 ,wefindthatr = √ 2. Using tan θ =b/a, we
find that tanθ = −1/ −1 = 1 so that you may be tempted to take θ = π/4. Figure 9.5
√shows the Argand diagram and it is clear that θ = −3π/4. Therefore,z = −1 − j =
2̸ −3π/4,and we see the importance ofdrawing an Argand diagram.
y
–1
2
5p —4
3p —4 –
–1
x
Figure9.5
Argand diagramdepictingz = −1 −jin Example 9.12.