Understanding Engineering Mathematics

We have

(^) (θ 1 ) (^) (θ 2 )=(cosθ 1 +jsinθ 1 )(cosθ 2 +jsinθ 2 )
=cosθ 1 cosθ 2 −sinθ 1 sinθ 2 +j(sinθ 1 cosθ 2 +sinθ 2 cosθ 1 )
=cos(θ 1 +θ 2 )+jsin(θ 1 +θ 2 )=^ (θ 1 +θ 2 )
from which it follows directly that
r 1 (θ 1 )r 2 (θ 2 )=r 1 r 2 (θ 1 +θ 2 )
So, to multiply in polar form:

• Multiplymodulii to obtainmodulus of product


• Addarguments to obtainargument of product.

or:
|z 1 z 2 |=|z 1 ||z 2 |
arg(z 1 z 2 )=arg(z 1 )+arg(z 2 )
Arg(z 1 z 2 )=Arg(z 1 )+Arg(z 2 )+ 2 kπ (k=any integer)

Note carefully the last result. The 2kπis needed because the principal value of the argu-
ments may add to give a value outside the principal value range.

Problem 12.8

Put

√

3 −jand 1−

√

3 jin polar form. Work the product of these both

in Cartesian and polar form and compare the results.

It helps to plot the numbers on the Argand plane – see Figure 12.2.

y

x

0

p/3

1 −√ 3 j

1

2

y

x

0

p/6

√ 3

(^2) √ 3
1
√ 3 − j
(i)
(ii)
Figure 12.2The complex numbers
√
3 −jand 1−
√
3 j.
We have|
√
3 −j|=2andArg(
√
3 −j)=−
π
6
,so
√
3 −j= 2 (−
π
6
).
Similarly,| 1 −
√
3 j|=2andArg( 1 −
√
3 j)=−
π
3
,so1−
√
3 j= 2
(
−
π
3
)
.