Division in polar form involves dividing two complex numbers that are represented in polar form. Polar form expresses a complex number in terms of its magnitude (modulus) and argument (angle). The polar form of a complex number
z is given by:
=
⋅
c
i
s
(
)
z=r⋅cis(θ)
Where:
r is the magnitude (distance from the origin to the complex number
z in the complex plane).
θ is the argument (angle formed between the positive real axis and the line connecting the origin and
z).
c
i
s
(
)
cis(θ) represents the complex exponential
c
i
s
(
)
=
cos
(
)
+
⋅
sin
(
)
cis(θ)=cos(θ)+i⋅sin(θ), where
i is the imaginary unit.
When dividing two complex numbers in polar form, you divide their magnitudes and subtract their arguments. Let's say you have two complex numbers
1
z
1
and
2
z
2
in polar form:
1
=
1
⋅
c
i
s
(
1
)
z
1
=r
1
⋅cis(θ
1
)
2
=
2
⋅
c
i
s
(
2
)
z
2
=r
2
⋅cis(θ
2
)
The division
1
2
z
2
z
1
can be calculated as:
1
2
=
1
⋅
c
i
s
(
1
)
2
⋅
c
i
s
(
2
)
=
1
2
⋅
c
i
s
(
1
−
2
)
z
2
z
1
=
r
2
⋅cis(θ
2
)
r
1
⋅cis(θ
1
)
=
r
2
r
1
⋅cis(θ
1
−θ
2
)
Where:
1
2
r
2
r
1
is the ratio of the magnitudes of the two complex numbers.
1
−
2
θ
1
−θ
2
is the difference of the arguments of the two complex numbers.
This result gives you a complex number in polar form. You can then express it in rectangular form (standard
+
a+bi form) if needed.
Keep in mind that when calculating the difference of the arguments
1
−
2
θ
1
−θ
2
, you need to be careful about the range of the angles. The result should be within the range
−
−π to
π to ensure that the angle remains the principal argument of the complex number.
In summary, to perform division of complex numbers in polar form:
Divide the magnitudes of the two complex numbers.
Subtract the argument of the denominator from the argument of the numerator.
Express the result in polar form by combining the magnitude and the adjusted argument.