axis when z is interpreted as a radius vector. The angle θ has an infinitely many possible values , including negative ones that differ by integral multiples of p . Those values can be determined from the equation tan θ = y where the quadrant containing the point corresponding to z must be specified. Each value of q is called an argument of z , and the set of all such values is obtained by adding multiple of p to q , and it is denoted by arg z .
Case (ii) There is a unique value of θ which satisfies the condition −< This value is called a principal value of θ or principal argument of z and is denoted by Arg z . Note that, Arg( ) z or Fig. . III-Quadrant θ = IV-Quadrant θ = θ = I-Quadrant π − θ = II-Quadrant π − θ = Principal Argument of a complex number O O O O θ = Fig.
. Fig. . Re Im z = r (cos θ + i sin θ ) o - - Complex Numbers The capital A is important here to distinguish the principal value from the general value.
Evidently, in practice to find the principal angle θ , we usually compute tan y x and adjust for the quadrant problem by adding or subtracting with appropriately. arg Arg z Some of the properties of arguments are ( ) arg arg arg z z ( ) arg arg arg ( ) arg arg ( ) The alternate forms of cos q q + i are cos( sin( ), ∈ . For instance the principal argument and argument of , , i − , and − i are shown below:- − − i Arg z ( ) − p arg z n p n p n p n p Illustration Plot the following complex numbers in complex plane (i) (ii) (iii) (iv) i sin