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

9.10 Loci and regions of the complex plane 351
6 √ 5̸ π/4+nπ/2 n=0,1,2,3
7 1̸ π/10 +2nπ/5 n = 0,1,2,3,4
10 16cos 5 θ−20cos 3 θ+5cosθ
11 16sin 5 θ−20sin 3 θ+5sinθ
9.10 LOCIANDREGIONSOFTHECOMPLEXPLANE
Regionsofthecomplexplanecanoftenbeconvenientlydescribedbymeansofcomplex
numbers. For example, the points that lie on a circle of radius 2 centred at the origin
(Figure 9.20) represent complex numbers all of which have a modulus of 2. The argumentsareanyvalueof
θ, −π < θ π.Wecandescribeallthepointsonthiscircleby
the simple expression
|z|=2
thatis,allcomplexnumberswithmodulus2.Wesaythatthelocus(orpath)ofthepoint
z is a circle, radius 2, centred at the origin. The interior of the circle is described by
|z| < 2 whileitsexterior isdescribed by |z| > 2.
Similarlyallpointslyinginthefirstquadrant(shadedinFigure9.21)havearguments
between 0 and π/2.This quadrantistherefore describedby the expression:
0 < arg(z) < π/2
y
y
2
z
y
2
x
x
p –4
arg z = p – 4
x
Figure9.20
A circle, radius2, centred atthe
origin.
Figure9.21
First quadrant ofthex--y plane.
Figure9.22
Locusofpointssatisfying
arg(z) = π/4.
Example9.23 Sketch the locus ofthe pointsatisfyingarg(z) = π/4.
Solution The set of points with arg(z) = π/4 comprises complex numbers whose argument is
π/4.Allthesecomplex numbers lie onthe lineshown inFigure 9.22.
Example9.24 Sketch the locus ofthe pointsatisfying |z −2| = 3.
Solution First mark the fixed point 2 on the Argand diagram labelling it ‘A’ (Figure 9.23). Consider
the complex numberzrepresented by the point P. From the vector triangle law of
addition
⃗OA + ⃗ AP = ⃗ OP
⃗AP = ⃗ OP − ⃗ OA