. . Polar form of a complex number Polar coordinates form another set of parameters that characterize the vector from the origin to the point z iy , with magnitude and direction. The polar coordinate system consists of a fixed point O called the pole and the horizontal half line emerging from the pole called the initial line (polar axis).
If r is the distance from the pole to a point P and q is an angle of inclination measured from the initial line in the counter clockwise direction to the line OP , then r and q of the ordered pair ( , ) r θ are called the polar coordinates of P . Superimposing this polar coordinate system on the rectangular coordinate system, as shown in diagram, leads to Fig. . Fig.
. Fig. . x = r cos θ ...( ) y = r sin θ .
...( ) Any non-zero complex number z iy can be expressed as z i r sin . Polar coordinates P ( r, ) O P ( x,y ) M Superimpose polar coordinates on rectangular coordinates O P ( x,y ) x+iy Rectangular coordinates O = cos θ = sin θ - - Definition . Let r and θ be polar coordinates of the point P x y ( , ) that corresponds to a non-zero complex number z iy . The polar form or trigonometric form of a complex number P is (cos sin ) For convenience, we can write polar form as iy r cis The value r represents the absolute value or modulus of the complex number z .
The angle θ is called the argument or amplitude of the complex number z denoted by arg (i) If z = , the argument θ is undefined; and so it is understood that z ¹ whenever polar coordinates are used. (ii) If the complex number z iy has polar coordinates( , ) r θ , its conjugate z iy has polar coordinates( , r . Squaring and adding ( ) and ( ), and taking square root, the value of r is given by r . Dividing ( ) by ( ), r x .
Case (i) The real number θ represents the angle, measured in radians, that z makes with the positive real