Question

Define the polar form of complex number.

Answer 1

Let the complex number z = x + iy be represented by the point P(x,y) in the complex plane.

Let |XOP| = θ and |OP| = r > 0

Then P(r, θ) are called the polar coordinates of P. 

where x = rcosθ 

y = rsinθ 

∴ z = r(cosθ+ i sin θ) 

This is called the polar form or trigonometric form or modulus – amplitude form of z. 

Keen Eye;

• r = √(x² + y²) = |z|  is called the modulus of z and 0 is called the argument (or amplitude) of z, written as arg (z) or amp (z). 
• The value of such that -π<θ≤n is called principal argument of z. 
• To find θ 

Case (i): 

When z is purely real. Then, it lies on the x-axis. 

(i) If x > 0, then θ=0 

(ii) If x < 0, then θ = π 

Case (ii): 

When z is purely imaginary. Then, it lies on the y-axis

Let tan α =|tan θ|

where 0 > α < π/2

(i) θ = α, when z lies in I quadrant 

(ii) θ – π – α, when z lies in II quadrant 

(iii) θ = α – π, when z lies in III quadrant 

(iv) θ = -α, when z lies in IV quadrant