### Mathematics Class XI

Unit-I: Sets and Functions
Chapter 1: Sets
Unit-II: Algebra
Chapter 5: Binomial Theorem
Chapter 6: Sequence and Series
Unit-III: Coordinate Geometry
Chapter 1: Straight Lines
Chapter 2: Conic Sections
Unit-IV: Calculus
Unit-V: Mathematical Reasoning
Unit-VI: Statistics and Probability
Chapter 1: Statistics
Chapter 2: Probability

# Polar representation of Complex Numbers

Argand Plane / Complex Plane:

The plane that has a complex number assigned to each of its points is called the Complex plane or Argand Plane.

The x-axis and the y-axis in the Argand plane are called,respectively the real axis and the imaginary axis.

The identification of a complex number on a plane was proposed by Jean-Robert Argand (1768-1822), hence the complex plane is known as the Argand plane.

Polar Representation of a Complex Number

Let the point have a polar coordinates .

Since has a cartesian we have:

–(i)

Clearly, —(ii)

Now, —(iii)

For any value of (i), (ii) and (iii) holds is called an Argument of denoted by , when , is not defined.

A unique value to by restricting to the interval i.e., , then it is called the Principal Argument.

is determined as follows:

where

Example:

Find

Solution:

Some Properties in Polar Form:

(i)

(ii)

If and are the two complex numbers expressed by:

and

then

and

If , we get:

(i)

(ii)

(iii)

(iv)

NOTE:

If , where is an integer or zero.

If , then graphical representation of

Example 1:

Express in the form of .

Solution:

Example 2:

If and

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