Find the principal argument of (1+i3 )^2. - Vidyadip

Question

Find the principal argument of $\Big(1+\text{i}\sqrt{3}\Big)^2.$

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Answer

Let $\text{z}=\Big(1+\text{i}\sqrt{3}\Big)^2$ $=1+3\text{i}^2+2\sqrt{3}\text{i}$ $=1-3+2\sqrt{3}\text{i}$ $=-2+2\sqrt{3}\text{i}$ Let $\beta$ be an acute angle given by $\tan\beta=\Big|\frac{\text{Im(z)}}{\text{Re(z)}}\Big|.$ Then, $\tan\beta=\Big|\frac{|2\sqrt{3}|}{|2|}\Big|=\big|\sqrt{3}\big|$ $\Rightarrow\tan\beta=\big|\tan\frac{\pi}{3}\big|$ $\Rightarrow\beta=\frac{\pi}{3}$ Clearly, z lies in the second quadrant. Therefore, the argument of z is given by $\text{arg(z)}=\pi-\frac{\pi}{3}=\frac{2\pi}{3}$ Hence, the principal argument of z is $\frac{2\pi}{3}.$

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