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

Solution:

The given complex number is,

z = √3 + i = r (cosθ + i sinθ) (Polar form) 

Let r cosθ = √3 and r sinθ = 1

On squaring and adding, we obtain

r² cos² θ + r² sin² θ = (√3)² + 1²

⇒ r² (cos² θ + sin² θ) = 3 + 1

⇒ r² = 4

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

Therefore,

2 cosθ = √3 and 2sinθ = 1

⇒ cosθ = √3/2 and sinθ = 1/2

Since, θ lies in quadrant I, θ = π/6

Hence,

√3 + i = r cosθ + ir sinθ

= 2 cos π/6 + i 2sin π/6

= 2 (cos π/6 + i sin π/6)

Thus, this is the required polar form.

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NCERT Solutions Class 11 Maths Chapter 5 Exercise 5.2 Question 7

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

Summary:

A complex number √3 + i is given. We have found its polar form to be 2 (cos π/6 + i sin π/6)