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 = 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 + 1²

⇒ 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 II, θ = π - π/4 = 3π/4

Hence,

- 1 + i = r cosθ + ir sinθ

= √2 cos 3π/4 + i √2 sin 3π/4

Thus, this is the required polar form.

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NCERT Solutions Class 11 Maths Chapter 5 Exercise 5.2 Question 4

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 that the polar form of - 1 + i to be 2 cos 3π/4 + i √2 sin 3π/4.