Question

Find the modulus and argument of mentioned below complex number and hence express in polar form:

$\sqrt{3}+i$

Answer 1

Let $z=1+i$

$|z|=|\sqrt{3}+i|$

$\Rightarrow |z|=\sqrt{(3{)}^2+1^2}$

$\Rightarrow |z|=\sqrt{2}\dots \left(1\right)$

Let $\alpha$ be acute angle such that

$\tan \alpha =\frac{|\text{Im}\left(z\right)|}{|\text{Re}\left(z\right)|}$

$\Rightarrow \tan \alpha =\frac{|1|}{|\sqrt{3}|}$

$\Rightarrow \alpha =\frac{\pi }{6}$

Since, $z$ lies in the first quadrant,

$\therefore arg\left(z\right)=\alpha$

$arg\left(z\right)=\frac{\pi }{6}$

Hence, Polar form of $z$

$z=|z|\left[\cos arg\right(z)+i\sin arg(z\left)\right]$

$\Rightarrow z=\sqrt{2}(\cos \frac{\pi }{6}+i\sin \frac{\pi }{6})$

Therefore, Polar form of $(\sqrt{3}+i)$ is $2(\cos \frac{\pi }{6}+i\sin \frac{\pi }{6})$