Question 2:

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

Answer

z = −√3 + i

Let r cos θ = −√3 and r sin θ = 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

∴ 2 cos θ = −√3 and 2 sin θ = 1

⇒ cos θ = −√3/2 and sin θ = 1/2

 

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

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