Q. 23.7( 92 Votes )

# Find the modulus and the arguments of each of the complex numbers in Exercises 1 to 2.

z = – √3 + i

Answer :

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 = r^{2} cos^{2} θ and 1 = r^{2} sin^{2} θ

Adding both the equations, we get

3 + 1 = r^{2} cos^{2} θ + r^{2} sin^{2} θ

⇒ 4 = r^{2} (cos^{2} θ + sin^{2} θ)

⇒ 4 = r^{2} or r^{2} = 4 [∵ sin^{2} θ + cos^{2} 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

Rate this question :

Find the modulus of each of the following complex numbers and hence express each of them in polar form: 1 – i

RS Aggarwal - MathematicsFind the modulus of each of the following complex numbers and hence express each of them in polar form: –i

RS Aggarwal - MathematicsFind the modulus of each of the following complex numbers and hence express each of them in polar form:

RS Aggarwal - MathematicsFind the modulus of each of the following complex numbers and hence express each of them in polar form:

RS Aggarwal - MathematicsFind the modulus of each of the following complex numbers and hence express each of them in polar form:

RS Aggarwal - MathematicsFind the modulus of each of the following complex numbers and hence express each of them in polar form: –1 + i

RS Aggarwal - MathematicsWrite 2i in polar form.

RS Aggarwal - MathematicsFind the principal argument of (–2i).

RS Aggarwal - Mathematics