Profile
Settings
Refer your friends
Sign out
Any number of the form a+bi is known as a Complex number, where a is the real part and b is the imaginary part of the given complex number. Value if i(iota) is √-1. Complex numbers can further be subcategorized into purely real(if b=0), purely imaginary(if a=0), Imaginary(If a and b are not equal to zero) depending on the properties they consist of. Zero is a number that is both purely real and purely imaginary. Two complex numbers are equal if their real parts are equal to each other and similarly, their imaginary parts should also be equal to each other for them to be equal complex numbers. A complex number can be represented in three forms. These three forms are Algebraic form, Polar form, and Exponential form(or Euler form). The polar form is also known as trigonometric form. In this article, we will discuss in detail the Polar Form of Complex Numbers.
Polar Form of Complex Number:
The polar form of any given Complex Number is written as below:
Z = r (Cos Θ + i Sin Θ)
Where r is the modulus of Z and Θ is the angle between the positive real axis and the line joining the point of origin.
r= |z| = √a2 +b2
Θ = arg(Z) = tan-1(y/x)
Representation and Derivation of Polar form(Trigonometric form) of Complex number:
Let us assume we have given complex number as:
Z = x+yi
Where
r= |z| {Modulus of complex number }
and Θ = arg(Z) {argument of complex number}
See the figure below:
Now, we can say that
x= r Cos Θ
And
y= r Sin Θ
Putting the values of x and y is a given complex number, we will get the polar form of the given complex number as below:
Z = x+yi
Z = r Cos Θ + i(r Sin Θ)
Z = r (Cos Θ + i Sin Θ)
Now this form of the complex number, Z = r (Cos Θ + i Sin Θ), is known as the Standard Polar form of Complex number or the trigonometric form of a Complex number.
We can write (Cos Θ + i Sin Θ) in the short form as, Cis Θ, where C represents Cos Θ, i represents iota, and s represents Sin Θ.
Therefore, Cis Θ = (Cos Θ + i Sin Θ)
Z = r (Cos Θ + i Sin Θ)
Putting the value of Cis Θ in Z = r (Cos Θ + i Sin Θ) above:
Z = r Cis Θ
Points to Remember:
- This form is known as trigonometric form as after conversion the complex number has trigonometric terms in it.
- And, it is known as a polar form because r and Θ are the polar coordinates of a point. Every point on the given plane will have one unique pair of (r, Θ).
- (x,y) are the Cartesian coordinates, and (r, Θ) are known as the polar coordinates. As we have made the conversion of the given complex number ‘Z = x+yi’ in the form of polar coordinates, the resulting form is called the Polar Form of the Complex Number.
- Also note that the ‘Z = x+yi’ form is known as the Algebraic form of complex numbers.
Let us see an example to understand the Polar form of Complex Number practically:
Example:
Convert Z = -1- √3i into its polar form.
Solution:
Z = -1- √3i
First, we need to find out r and Θ, to convert the given complex number into its form.
r= √(-1)2 +(–V3)2
= 2
tan α = √3
α is in 3rd quadrant so
α =4π/3
Θ =4π/3
As, we have found out the values of r and Θ, we can now put those values in the standard polar form of complex number to get the polar form of the given complex number.
Therefore, polar form of given complex number (Z = -1- √3i) will be:
Z = r (Cos Θ + i Sin Θ)
Z = 2 (Cos(4π/3) + iSin (4π/3))
Conclusion:
In this article we discussed the polar form of complex numbers in detail. We already know that Complex numbers are a combination of real numbers and imaginary numbers. We saw that complex numbers can be represented in three forms and the form in which polar coordinates or trigonometric functions are used to write the expression, that form is known as Polar form or Trigonometric form of Complex numbers. We learnt derivation of formulas and tried some examples for practical implication of the formulas derived above. We hope this study material would be helpful for you to clear your basic concepts on the given topic.
