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If rectangular form of complex number is shown as \(z = \frac{5}{2} + \frac{{5\sqrt 3 }}{2}i\) then its polar form is represented as – 
1. \(5\left( {\cos \left( {\frac{{2\pi }}{3}} \right) - isin\left( {\frac{{2\pi }}{3}} \right)} \right)\)
2. \(5\left( {\cos \left( {\frac{\pi }{3}} \right) - isin\left( {\frac{\pi }{3}} \right)} \right)\)
3. \(5\left( {\cos \left( {\frac{{2\pi }}{3}} \right) + isin\left( {\frac{{2\pi }}{3}} \right)} \right)\)
4. \(5\left( {\cos \left( {\frac{\pi }{3}} \right) + isin\left( {\frac{\pi }{3}} \right)} \right)\)

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Correct Answer - Option 4 : \(5\left( {\cos \left( {\frac{\pi }{3}} \right) + isin\left( {\frac{\pi }{3}} \right)} \right)\)

CONCEPT:

Point P is uniquely determined by the ordered pair of real numbers (r, θ), called the polar coordinates of the point P.

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

CALCULATION:

Given complex number is \(z = \frac{5}{2} + \frac{{5\sqrt 3 }}{2}i\)

\(rcos\theta = \frac{5}{2},\;rsin\theta = \frac{{5\sqrt 3 }}{2}\)

By squaring and adding, we get –

\({{\rm{r}}^2}\left( {{\rm{co}}{{\rm{s}}^2}{\rm{\theta }} + {\rm{si}}{{\rm{n}}^2}{\rm{\theta }}} \right) = \frac{{100}}{4} = 25\)

∴ r = 5

\( \Rightarrow cos\theta = \frac{{\frac{5}{2}}}{r} = \frac{{\frac{5}{2}}}{5} = \frac{1}{2}\) and \(sin\theta = \frac{{\frac{{5\sqrt 3 }}{2}}}{r} = \frac{{\frac{{5\sqrt 3 }}{2}}}{5} = \frac{{\sqrt 3 }}{2}\)

Since it is in first quadrant, \(\theta = \frac{\pi }{3}\)

So, on comparing with z = r (cosθ + i sinθ), we can write as \(5\left( {\cos \left( {\frac{\pi }{3}} \right) + isin\left( {\frac{\pi }{3}} \right)} \right)\) 

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