Find the modulus and arguments of each of the complex numbers in Exercises 1 to 2:
  z = - 1 - i√3

Solution:

The given complex number is,

z = - 1 - i√3

Let r cosθ = - 1 and r sinθ = - √3

On squaring and adding, we obtain

(r cosθ)² + (r sinθ)² = (- 1)² + (- √3)²

⇒ r² (cos² θ + sin² θ) = 1+ 3 [∵ cos² θ + sin² θ = 1]

⇒ r² = 4

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

Therefore, Modulus = 2

Hence, 2cosθ = - 1 and 2sinθ = - √3

⇒ cosθ = - 1/2 and sinθ = - √3/2

Since both the values of sinθ and cosθ are negative in III quadrant,

Argument, θ =- 2π/3

Thus, the modulus and argument of the complex number - 1 - i√3 are 2 and - 2π/3 respectively.

---

NCERT Solutions Class 11 Maths Chapter 5 Exercise 5.2 Question 1

Find the modulus and arguments of each of the complex numbers in Exercises 1 to 2:  z = - 1 - i√3

Summary:

A complex number is given. We have found that the modulus and argument of the complex number - 1 - i√3 are 2 and - 2π/3 respectively