Question

Find the principal arguments of the following Complex Number z = 29 + 29i.
1. π / 4
2. π / 2
3. π / 3
4. π / 6

Answer 1

Correct Answer - Option 1 : π / 4

CONCEPT:

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 = 29 + 29i.

As it can be clearly seen it lies in first quadrant.

∴ Principle argument \({\rm{\theta }} = {\tan ^{ - 1}}\frac{{\rm{y}}}{{\rm{x}}}\)

⇒ \({\rm{\theta }} = {\tan ^{ - 1}}\frac{{29}}{{29}} = {\tan ^{ - 1}}1 = \frac{{\rm{\pi }}}{4}\)