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

9.5.1 Multiplicationanddivisioninpolarform
9.5 Polar form of a complex number 335
ThepolarformmayseemmorecomplicatedthantheCartesianformbutitisoftenmore
useful. For example, suppose wewant tomultiplythe complex numbers
We find
z 1
=r 1
(cosθ 1
+jsinθ 1
) and z 2
=r 2
(cosθ 2
+jsinθ 2
)
z 1
z 2
=r 1
(cosθ 1
+jsinθ 1
)r 2
(cosθ 2
+jsinθ 2
)
=r 1
r 2
{(cosθ 1
cosθ 2
−sinθ 1
sinθ 2
) +j(sinθ 1
cosθ 2
+sinθ 2
cosθ 1
)}
which can be written as
r 1
r 2
{cos(θ 1
+ θ 2
) +jsin(θ 1
+ θ 2
)}
using the trigonometric identities of Section 3.6. This is a new complex number which,
ifwecomparewiththegeneralformr(cosθ +jsin θ),weseehasamodulusofr 1
r 2
and
an argument of θ 1
+ θ 2
. To summarize: to multiply two complex numbers we multiply
their moduliand add their arguments, thatis
z 1
z 2
=r 1
r 2̸ (θ 1
+θ 2
)
Example9.13 Ifz 1
= 3̸ π/3 andz 2
= 4̸ π/6 findz 1
z 2
.
Solution Multiplyingthemoduliwefindr 1
r 2
= 12,andaddingtheargumentswefind θ 1
+ θ 2
=
π/2.Thereforez 1
z 2
= 12̸ π/2.
Asimilardevelopmentshowsthattodividetwocomplexnumberswedividetheirmoduli
and subtracttheirarguments,that is
z 1
z 2
= r 1
r 2
̸ (θ 1
−θ 2
)
Example9.14 Ifz 1
= 3̸ π/3 andz 2
= 4̸ π/6 findz 1
/z 2
.
Solution Dividing the respective moduli, we find r 1
/r 2
= 3/4 and subtracting the arguments,
π/3 − π/6 = π/6.Hencez 1
/z 2
= 0.75̸ π/6.
EXERCISES9.5
1 Markonan Arganddiagrampointsrepresenting
z 1 =3−2j,z 2 =−j,z 3 =j 2 ,z 4 =−2−4jand
z 5 = 3. Find the modulus andargumentofeach
complex number.
2 Expressthe following complex numbersin polar
form:
(a) 3 −j (b) 2 (c) −j (d) −5 +12j
3 Find the modulusandargumentof(a)z 1 = − √ 3 +j
and (b)z 2 = 4 +4j. Hence expressz 1 z 2 andz 1 /z 2 in
polar form.
4 Express √ 2̸ π/4,2̸ π/6and2̸ −π/6 in Cartesian
forma +bj.
5 Prove the result z 1
z 2
= r 1
r 2
̸ θ 1 −θ 2 .