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

9.5 Polar form of a complex number 333
Solutions
1 SeeFigure S.17.
2 (a) SeeFigure S.18. (b) j(1 +j) = −1 +j
Imaginary
Imaginary
–2
7+2j
–1+j 1
1+j
–0.5j
3
Real
–3–3j
FigureS.17
–1
FigureS.18
1
Real
9.5 POLARFORMOFACOMPLEXNUMBER
ItisoftenusefultoexchangeCartesiancoordinates (a,b)forpolarcoordinatesrand
θ asdepictedinFigure 9.3.
FromFigure 9.3 wenotethat
cosθ = a r
sinθ = b r
and so,
a=rcosθ
b=rsinθ
Furthermore,
tanθ = b a
UsingPythagoras’stheoremweobtainr = √ a 2 +b 2 .Byfindingrand θ wecanexpress
the complex numberz =a +bj in polar formas
z=rcosθ+jrsinθ =r(cosθ+jsinθ)
whichweoftenabbreviatetoz =r̸ θ.Clearly,risthe‘distance’ofthepoint (a,b)from
the origin and is called the modulus of the complex numberz. The modulus is always
a non-negative number and is denoted |z|. The angle is conventionally measured from
the positivexaxis. Angles measured in an anticlockwise sense are regarded as positive
whilethosemeasuredinaclockwisesenseareregardedasnegative.Theangle θ iscalled
the argument ofz, denoted arg(z). Since adding or subtracting multiples of 2π from θ
will result in the ‘arm’ in Figure 9.3 being in the same position, the argument can have
many values. Usuallywe shall choose θ tosatisfy −π < θ π.
Cartesian form:z =a +bj
Polarform:z =r(cosθ +jsinθ) =r̸
|z|=r= √ a 2 +b 2
θ
a=rcosθ b=rsinθ tanθ = b a