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Given two complex numbers \({Z_1} = 5 + \left( {5\sqrt 3 } \right)i,\) and \({Z_2} = \frac{2}{{\surd 3}} + 2i\) the argument of \(\frac{{{Z_1}}}{{{Z_2}}}\) in degrees is

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Correct Answer - Option 1 : 0

Explanation:

The argument of a complex number z = x + iy

\(\arg z = {\tan ^{ - 1}}\left( {\frac{y}{x}} \right)\)

\({Z_1} = 5 + \left( {5\sqrt 3 } \right)i,{Z_2} = \frac{2}{{\surd 3}} + 2i\)

Then \(\arg{Z_1} = {\tan ^{ - 1}}\left( {\frac{{5\sqrt 3 }}{5}} \right) = 60^\circ\)

And \(\arg{Z_2} = {\tan ^{ - 1}}\left( {\frac{2}{{\frac{2}{{\sqrt 3 }}}}} \right) = 60^\circ\)

So, \(\arg \left( {\frac{{{Z_1}}}{{{Z_2}}}} \right) = \arg \left( {{Z_1}} \right) - {\rm{arg}}\left( {{Z_2}} \right)\)

\(\arg \left( {\frac{{{Z_1}}}{{{Z_2}}}} \right) \) = 60° - 60° = 0

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