Convert each of the complex numbers given in Exercises 3 to 8 in the polar form: - 1 - i

Solution:

The given complex number is,

z = -1 - i

Let z = - 1 - i = r (cosθ + i sinθ) (Polar form) 

Let r cosθ = - 1 and r sinθ = - 1

On squaring and adding, we obtain

r² cos² θ + r² sin² θ = (- 1)² + (- 1)²

⇒ r² (cos² θ + sin² θ) = 1 + 1

⇒ r² = 2

⇒ r = √2 [∵ Conventionally, r > 0]

Therefore,

√2 cosθ = - 1 and √2 sinθ = - 1

⇒ cosθ = - 1/√2 and sinθ = - 1/√2

Since, θ lies in the quadrant III, θ = π + π/4 = 5π/4 (or) 5π/4 - 2π = -π/4 (Because adding or subtracting 2π to an angle does not make any difference)

Hence,

- 1 - i = r cosθ + ir sinθ

= √2 cos (-π/4) + i√2 sin (-π/4)

Thus, this is the required polar form.

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NCERT Solutions Class 11 Maths Chapter 5 Exercise 5.2 Question 5

Convert each of the complex numbers given in Exercises 3 to 8 in the polar form: - 1 - i

Summary:

A complex number - 1 - i is given. We have found its polar form to be √2 cos (-π/4) + i√2 sin (-π/4).