Find the modulus and the argument of the complex number z = -√3 + i

z = -√3 + i

Let rcosθ = -√3 and rsinθ = 1

On squaring and adding, we obtain

r² cos² θ + r² sin² θ = (-√3)² + 1²

⇒ r² = 3 + 1 = 4                          [cos² θ + sin² θ = 1]

⇒ r = √4 = 2                                [Conventionally, r > 0]

∴ Modulus = 2

∴ 2cosθ = -√3 and 2sinθ = 1

⇒ cosθ = -^√3/₂ and sinθ = ¹/₂

∴ θ = π - ^π/₆ = ^5π/₆   [As θ lies in the II quadrant]

Thus, the modulus and argument of the complex number -√3 + i are 2 and ^5π/₆ respectively.