# Find the modulus and the arguments of each of the complex numbers in Exercises 1 to 2.z = – √3 + i

As we know that the polar representation of a complex number z = x + iy is

z = r (cos θ + i sin θ) where is the modulus of the complex number and θ is the argument of the complex number, denoted by arg z.

So, now, let –√3 = r cos θ and 1 = r sin θ ……….(i)

Squaring both sides, we get

3 = r2 cos2 θ and 1 = r2 sin2 θ

Adding both the equations, we get

3 + 1 = r2 cos2 θ + r2 sin2 θ

4 = r2 (cos2 θ + sin2 θ)

4 = r2 or r2 = 4 [ sin2 θ + cos2 θ = 1]

r = 4

r = 2 (conventionally, r>0) ……….(ii)

Substituting r = 2 in (i), we get

–√3 = 2 cos θ and 1 = 2 sin θ

We know that the complex number –√3 + i lies in the second quadrant and the value of the argument lies between - π and π, i.e. - π < θ ≤ π.

……….(iii)

From (ii) and (iii), we have

r = 2 and

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Write 2i in polar form.

RS Aggarwal - Mathematics