Polar Form Of Complex Numbers

The polar form of complex numbers is simply the alternative way of expressing complex numbers from their standard form Z = a + bi. The standard form is also known as the rectangular form, and this is one of the most popular expressions in Complex numbers. However, the polar form of expression becomes very relevant when connecting the complex number to trigonometry.

The polar form of complex numbers expression includes the cosϴ and sinϴ. Hence it is very important to understand trigonometry. This article will consider the polar form of a complex number in five aspects

How to convert the standard form z = a + bi to polar form

Converting from degree to radian and vice versa

How to multiply the two polar forms of complex numbers

How to divide two polar forms of complex numbers.

The Polar Form of Complex Number Expression

The polar form of complex numbers is expressed as;

Z = Cosrϴ + isinrϴ

Now, if you compare the above form to the standard complex number form, you will see that

Z = Cosrϴ + isinrϴ is the same with Z = a + bi

Where,

a = cosrϴ

b = sinrϴ

Now the polar form Z = Cosrϴ + isinrϴ can also be expressed as

Z = r[Cosϴ + isinϴ].

The r in the above polar form is known as the modulus or absolute value of the complex number while the ϴ is the arguments of the complex number. While you can find more about the modulus and the arguments here, their respective formula is given as

r = √a² + b²

ϴ Can be expressed as any of the four depending on the quadrant of the complex number being converted. These are the four possible values of ϴ depending on the quadrants

The formula for the argument is given as

ϴ = ϴ_ref when in the first quadrant of the Argand diagram

ϴ = 180 – ϴ_ref when in the second quadrant of the Argand diagram

ϴ = 180 + ϴ_ref when in the third quadrant of the Argand diagram

ϴ = 360 – ϴ_ref when in the fourth quadrant of the Argand diagram

Where;

 ϴ_ref  is areference angle and is given as

In some instances, the reference angle could just be any of 0⁰, 90⁰,180⁰ or 270⁰. However, this only happens in special cases.

How to Change the Standard Form of a Complex Number to Polar Form

Example 1

Convert the standardized complex number Z = 3 + 3i to the polar form

Solution

The polar form of the complex us given as

Z = r[Cosϴ + isinϴ].

a = 3, b = 3

Now to find ϴ, we have to determine the quadrant of the reference angle in the argand diagram. We have treated how to determine this in our Argand Diagram and Modulus topics. The argand diagram and modulus will become something like this

Since a = positive and b = positive 3, they will be in the first quadrant of the Argand diagram.

ϴ = ϴ_ref when in the first quadrant of the Argand diagram

Therefore,  ϴ =45⁰

With the above,

Z = r[Cosϴ + isinϴ]

The above is the polar form of Z =3 + 3i

How to Convert a polar form from degree to radians and vice versa?

After finding the answer to a problem, you may be asked to leave your answer in radians. The process is very simple. All you need do is multiply your derived ϴ by π/180.

Now, if you are given a polar form without the degree sign, then it means it is in radians. To convert to a degree, you will multiply the value by 180/π

Example 2

Now change the Polar form in example 1 from degree to radians

Solution

The polar form is

To change to radians, we will have the following below

45⁰ x π/180 = π/4

As such, we have

Example 2

Solution

The a will be in the positive x axis while the b will be in the negative y axis

So our plotted diagram will be in the form below

The expression of the polar form for complex numbers is given as

Z = r[Cosϴ + isinϴ]

Where

Now For ϴ

Since the plotted angle is in the fourth quadrant, then;  

ϴ = 360 – ϴ_ref when in the third quadrant of the Argand diagram

= 60⁰

This is not the final answer as our interest is in getting the argument which is ϴ. As such

ϴ = 360⁰ – ϴ_ref

ϴ = 360⁰ – 60⁰ = 300⁰

With the above argument and modulus, the polar form of the complex number

Z = r[Cosϴ + isinϴ]

Z = 4[Cos300⁰ + isin300⁰]

Multiplication and Division of two Polar Complex Numbers

Example 8

Z1 = 5(cos30⁰ + isin30⁰)

Z2 = 5(cos30⁰ + isin30⁰)

Multiply z1 and z2

Solution

Step 1: multiply the modulus of the two complex numbers. Then add their respective degrees together with the degree of the cosines in the z1 and Z2 multiplying each other and the sines in z1 and z2 multiplying each other

Z1 x Z2 = 5 x 5 [ cos(30⁰ + 30⁰) + isin(30⁰ + 30⁰)] Z1 x Z2 =25 ( COS60⁰ + isin 60⁰)

Example 9: Divide the equation

Getting the division of the complex number is similar to finding the multiplication. The only difference is to subtract the modulus of the complex numbers as opposed to adding them together, which was done when computing multiplications.

Example

Let  Z1 = 6(cos45⁰ + isin45⁰) and Z2 = 2(cos30⁰ + isin30⁰). Find their division Z1/Z2

Solution

Z1/Z2 = r₁/r₂(cos(ϴ₁ – ϴ₂) + isin(ϴ₁– ϴ₂))

ϴ₁ = 45⁰, ϴ₂ = 30⁰, r₁ = 6, r₂ = 2

Z1/Z2 = 6/2(cos(45 – 30) + isin(45 – 30))

Z1/Z2 = 3(cos15⁰ + isin15⁰)

The multiplication and division method of the polar form is very similar. It is mostly about small changes in their signs. Once students can handle that, then they will not have an issue solving any problem handling similar challenges.

Conclusion

The polar form of the complex number is one of the more popular expressions in the complex plane hierarchy. The expression allows students successfully inculcate complex numbers into trigonometry. Understanding the polar form of a complex number requires you to know the standard form of the complex number and its algebra, the Argand diagram, the Modulus of complex numbers and finally, the Argument of complex numbers.

This article has successfully considered the algebra of the polar form and how they work. These algebras are very important for understanding the Euler rule and De Moivre Theorem, which will be considered in our next topic.