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 the given complex number.

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

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

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

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

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

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

CALCULATION:

Given that \(\left| z \right| = \sqrt 2 \) and angle with x-axis is 45°.

Since its in the third quadrant -

∴ \(\theta = - \pi + \frac{\pi }{4} = - \frac{{3\pi }}{4}\)

⇒ [r, θ] = \(\left[ {\sqrt 2 , - \frac{{3\pi }}{4}} \right]\)