Represent the complex number Z = - 2 - i 2√3 in the polar form.

CONCEPT:

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

Here \(r = \sqrt {{x^2} + {y^2}} = \left| z \right|\) is called modulus of given complex number.

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

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

Here \(r = \sqrt {{x^2} + {y^2}} = \left| z \right|\) is called modulus of given complex number.

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, \({\rm{\theta }} = {\tan ^{ - 1}}\frac{{\rm{y}}}{{\rm{x}}}\)

For second quadrant \({\rm{\theta }} = {\rm{\pi }} - {\tan ^{ - 1}}\frac{{\rm{y}}}{{\rm{x}}}\)

For third quadrant \({\rm{\theta }} = - {\rm{\pi }} + {\tan ^{ - 1}}\frac{{\rm{y}}}{{\rm{x}}}\)

For fourth quadrant, \({\rm{\theta }} = - {\rm{\;ta}}{{\rm{n}}^{ - 1}}\frac{{\rm{y}}}{{\rm{x}}}\)

CALCULATION:

Given complex number is \(Z = - 2 - i2\sqrt 3 \)

rcosθ = - 2, \(rsin\theta = - 2\sqrt 3 \)

By squaring and adding, we get:

\({{\rm{r}}^2}\left( {{\rm{co}}{{\rm{s}}^2}{\rm{\theta }} + {\rm{si}}{{\rm{n}}^2}{\rm{\theta }}} \right) = 4 + 12 \)

∴ r = 4

\(\Rightarrow cos\theta = \frac{{ - 2}}{r} = \frac{{ - 2}}{4} = - \frac{1}{2}\:and\:sin\theta = \frac{{ - 2\sqrt 3 }}{r} = \frac{{ - 2\sqrt 3 }}{4} = - \frac{{\sqrt 3 }}{2}\)

Since it is in third quadrant

\(\theta = - \pi + \frac{\pi }{3} = - \frac{{2\pi }}{3}\)

So, on comparing with z = r (cosθ + i sinθ),

we can write as

→  \(4\left( {\cos \left( { - \frac{{2\pi }}{3}} \right) + isin\left( { - \frac{{2\pi }}{3}} \right)} \right)i.e.4\left( {\cos \left( {\frac{{2\pi }}{3}} \right) - isin\left( {\frac{{2\pi }}{3}} \right)} \right)\).