Find the principal argument of (–2i).

1.	Find the principal argument of (–2i).
Answer» Let, z = -2i Let 0 = r cosθ and -2 = r sinθ By squaring and adding, we get (0)² + (-2)² = (r cosθ)² + (r sinθ)² ⇒ 0+4 = r²(cos²θ + sin²θ) ⇒ 4 = r² ⇒ r = 2 ∴ cosθ = 0 and sinθ = -1 Since, θ lies in fourth quadrant, we have θ = -π/2 Since, θ ∈ (-π ,π ] it is principal argument.

Answer»

Let, z = -2i

Let 0 = r cosθ and -2 = r sinθ

By squaring and adding, we get

(0)² + (-2)² = (r cosθ)² + (r sinθ)²

⇒ 0+4 = r²(cos²θ + sin²θ)

⇒ 4 = r²

⇒ r = 2

∴ cosθ = 0 and sinθ = -1

Since, θ lies in fourth quadrant, we have

θ = -π/2

Since, θ ∈ (-π ,π ] it is principal argument.

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