Correct Answer - Option 3 :
\(\frac{{2{\rm{\pi }}}}{3}\)
Given:
Complex number \(z = - 7 + 7\sqrt 3 i\)
CONCEPT:
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 \({\rm{\theta }} = {\rm{π }} - {\tan ^{ - 1}}\frac{{\rm{y}}}{{\rm{x}}}\)
For third quadrant \({\rm{\theta }} = - {\rm{π }} + {\tan ^{ - 1}}\frac{{\rm{y}}}{{\rm{x}}}\)
For fourth quadrant \({\rm{\theta }} = - {\tan ^{ - 1}}\frac{{\rm{y}}}{{\rm{x}}}\)
CALCULATION:
Given complex number is \(z = - 7 + 7\sqrt 3 i\).
As it can be clearly seen it lies in second quadrant.
⇒ Principle argument \({\rm{\theta }} = {\rm{π }} - {\tan ^{ - 1}}\frac{{\rm{y}}}{{\rm{x}}}\)
\( ⇒ {\rm{\theta }} = {\tan ^{ - 1}}\frac{{7\sqrt 3 }}{{ - 7}} = {\tan ^{ - 1}}\left( { - \sqrt 3 } \right) = {\rm{π }} - \frac{{\rm{π }}}{3} = \frac{{2{\rm{π }}}} {3}\)
∴ The value of the principal arguments is 2π/3.
