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  • Find principal argument ofbracket1plusiroot3square

Find principal argument of  (1+i\sqrt3)^2.

 

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\begin{array}{c} (1+i \sqrt{3})^{2}=1+i^{2} \cdot 3+2 \sqrt{3} i \\ =1-3+2 \sqrt{3} i=-2+2 \sqrt{3} i \\ \tan \alpha=\left|\frac{2 \sqrt{3}}{-2}\right| \\ \tan \alpha=|-\sqrt{3}|=\sqrt{3} \\ \tan \alpha=\tan \frac{\pi}{3} \\ \alpha=\frac{\pi}{3} \end{array}

 

  Now Re(z)<0 and image(z)>0 arg(z)=\pi-\alpha=\pi-\frac{\pi}{3}=\frac{2\pi}{3}

\text {Hence, the principal arg}=\frac{2 \pi}{3}

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