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Let z= \frac{\left ( -1-\sqrt{3}i \right )\left ( \sqrt{3}+i \right )}{\left ( 1-i \right )} then arg\left ( z \right ) equals 

  • Option 1)

    \frac{3\pi }{4}

  • Option 2)

    \frac{-5\pi }{4}

  • Option 3)

    \frac{7\pi }{4}

  • Option 4)

    \frac{\pi }{4}

 

Answers (1)

best_answer

arg(z)= \left \{ arg{(-1-\sqrt{3}i)(\sqrt3+i)} \right \}- arg \left ( 1-i \right )

arg(z)= \left \{ arg{(-1-\sqrt{3}i)\right\} +arg\left\{(\sqrt3+i)} \right \}- arg \left ( 1-i \right )

arg (-1-\sqrt3i)= \tan ^{-1}\left | \frac{-\sqrt{3}}{-1} \right |-\pi =\frac{\pi }{3} -\pi = \frac{-2\pi }{3}

arg (\sqrt3+i) = \tan ^{-1}\left | \frac{1}{\sqrt{3}} \right |= \frac{\pi }{6}

arg (1-i) = -\tan ^{-1}\left | \frac{-1}{1} \right |=-\frac{\pi }{4}

arg (z) = \frac{-2\pi }{3}+\frac{\pi }{6}+\frac{\pi }{4}+2n\pi

arg(z) = \frac{-\pi }{4}+ 2n\pi

for n= 1 we get arg(z) \frac{7\pi }{4}

 

Properties of Argument of a Complex Number -

Arg\left(\frac{z}{w}\right)=Arg(z)-Arg(w)+2n\pi

- wherein

n\epsilon Integer and it is chosen such that Arg(z/w) lies in principal value range of Argument.

 

 

 


Option 1)

\frac{3\pi }{4}

This is incorrect

Option 2)

\frac{-5\pi }{4}

This is incorrect

Option 3)

\frac{7\pi }{4}

This is correct

Option 4)

\frac{\pi }{4}

This is incorrect

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