Multiplication of complex numbers in polar form is quite straightforward. Complex numbers can be represented in polar form as
=
(
cos
β‘
+
sin
β‘
)
z=r(cosΞΈ+isinΞΈ), where
r is the magnitude or modulus of the complex number and
ΞΈ is the argument or angle of the complex number with respect to the positive real axis.
When you multiply two complex numbers in polar form, you simply multiply their magnitudes and add their arguments. In other words, if you have two complex numbers
1
=
1
(
cos
β‘
1
+
sin
β‘
1
)
z
1
β
=r
1
β
(cosΞΈ
1
β
+isinΞΈ
1
β
) and
2
=
2
(
cos
β‘
2
+
sin
β‘
2
)
z
2
β
=r
2
β
(cosΞΈ
2
β
+isinΞΈ
2
β
), then their product
1
β
2
z
1
β
β
z
2
β
in polar form is given by:
1
β
2
=
1
β
2
(
cos
β‘
(
1
+
2
)
+
sin
β‘
(
1
+
2
)
)
z
1
β
β
z
2
β
=r
1
β
β
r
2
β
(cos(ΞΈ
1
β
+ΞΈ
2
β
)+isin(ΞΈ
1
β
+ΞΈ
2
β
))
Here's a step-by-step breakdown of the process:
Multiply the magnitudes:
1
β
2
r
1
β
β
r
2
β
gives you the magnitude of the resulting complex number.
Add the arguments:
1
+
2
ΞΈ
1
β
+ΞΈ
2
β
gives you the angle of the resulting complex number.
Convert back to polar form: Use the magnitude and angle to represent the result in polar form.
Remember that the angle should be adjusted to be within the appropriate range (usually
β
βΟ to
Ο or
0
0 to
2
2Ο, depending on the conventions you're following).
Let's work through an example:
Suppose you have two complex numbers:
1
=
2
(
cos
β‘
4
+
sin
β‘
4
)
z
1
β
=2(cos
4
Ο
β
+isin
4
Ο
β
)
2
=
3
(
cos
β‘
3
+
sin
β‘
3
)
z
2
β
=3(cos
3
Ο
β
+isin
3
Ο
β
)
First, multiply the magnitudes:
1
β
2
=
2
β
3
=
6
r
1
β
β
r
2
β
=2β
3=6.
Then, add the arguments:
1
+
2
=
4
+
3
=
7
12
ΞΈ
1
β
+ΞΈ
2
β
=
4
Ο
β
+
3
Ο
β
=
12
7Ο
β
.
So, the product of
1
z
1
β
and
2
z
2
β
in polar form is:
1
β
2
=
6
(
cos
β‘
7
12
+
sin
β‘
7
12
)
z
1
β
β
z
2
β
=6(cos
12
7Ο
β
+isin
12
7Ο
β
)
That's how you multiply complex numbers in polar form!