Q.Find the modulus and argument of $z=1+i\sqrt3$ and write it in polar form.

Compute $r=\sqrt{a^{2}+b^{2}}$, locate the quadrant, then find $\theta$ from $\tan\theta=b/a$.

Here $a=1,\ b=\sqrt3$.

Modulus:

Argument: the point $(1,\sqrt3)$ has $a>0,\ b>0$, so it lies in Quadrant I and the principal argument equals the reference angle:

Polar form:

$z=r(\cos\theta+i\sin\theta)=2\left(\cos\frac{\pi}{3}+i\sin\frac{\pi}{3}\right).$ …

Writing $r=\sqrt{1+\sqrt3}$ by forgetting to square $\sqrt3$ — note $(\sqrt3)^{2}=3$. Also, always c …