Find the polar form of complex number Z = 3 + 3i.

  1. 3 (cos 45°+ i sin 45°)
  2. √2(cos 45° + i sin 45°)
  3. 3√2(cos 45° - i sin 45°)
  4. 3√2(cos 45° + i sin 45°)
  5. None of these

Answer (Detailed Solution Below)

Option 4 : 3√2(cos 45° + i sin 45°)
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Detailed Solution

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CONCEPT:

Modulus of complex number:

Let the point P represent the nonzero complex number z = x + iy.

Here  is called modulus of given complex number.

Argument of complex number:

For any complex number z ≠ 0, there corresponds only one value of θ in 0 ≤ θ < 2π. However, any other interval of length 2π, for example – π < θ ≤ π, can be such an interval. We shall take the value of θ such that – π < θ ≤ π, called principal argument of z and is denoted by arg (z).

The argument of Z is measured from positive x-axis only.

Let z = r (cosθ + i sinθ) is polar form of any complex number then following ways are used while writing θ for different quadrants –

For first quadrant

For second quadrant

For third quadrant 

For fourth quadrant

CALCULATION:

Given complex number Z = 3 + 3i.

 Let rcosθ = 3 and rsinθ = 3

By squaring and adding, we get

∴ r = 3√2

As we have taken rcosθ = 3

⇒ θ = 45° (Since it is in first quadrant, there won’t be any changes in θ value)

So, on comparing with z = r (cosθ + i sinθ), we can write as .

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