Let P (x, y) be any point in the argand plane corresponding to a complex number z = x + i y such that it makes an angle θ with positive direction of X-axis then the polar form of z = x + i y is-
z = r (cos θ + i sin θ) where r = | z | = √(x2 + y2) and tan θ = | y/x |
The argument of z depends upon the Quadrant in which the complex number lies.
Case I- If z lies in the first quadrant then the argument of (z) = θ.
Case II- If z lies in the second quadrant then the argument of (z) = 180° – θ.
Case III- If z lies in the third quadrant then the argument of (z) = 180° + θ or θ – 180°.
Case IV- If z lies in the fourth quadrant then the argument of (z) = 360° – θ or -θ.
Example- Express the following complex number in Polar Form √3 – i
Solution- Let z = √3 – i Now, r = | z | = √[(√3)2 + (-1)2] = √(3 + 1) = 2
and tan θ = | y/x | = | -1/√3 | = 1/√3 ∴ θ = 30°
Since z lies in the fourth quadrant, ∴ arg (z) = α = -θ = -30°
∴ The required polar form of z = √3 – i is, z = r [cos α + i sin α] ⇒ z = 2 [cos (-30°) + i sin (-30°)] ⇒ z = 2 [cos 30° – i sin 30°]
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