Question 1:
Find the modulus and the argument of the complex number z = −1 – i√3
Answer
z = −1 – i√3
Let r cos θ = −1 and r sin θ = −√3
On squaring and adding, we obtain
(r cos θ)2 + (r sin θ)2 = (−1)2 + (−√3)2
⇒ r2(cos2 θ + sin2 θ) = 1 + 3
⇒ r2 = 4 [cos2 θ + sin2 θ = 1]
⇒ r = √4 = 2 [Conventionally, r >0]
∴ Modulus = 2 ∴ 2 cos θ = −1 and 2 sin θ = −√3
⇒ cos θ = −1/2 and sin θ = −√3/2
Since both the values of sin θ and cos θ are negative and sin θ and cos θ are negative in III quadrant,
Thus, the modulus and argument of the complex number −1 – √3i are 2 and −2π/3 respectively.
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