Question

Represent $z=1+i \sqrt{3}$ in the polar form.

• A. $\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}$
• B. $\cos \frac{\pi}{3}-i \sin \frac{\pi}{3}$
• C. $2\left(\cos \frac{\pi}{3}+ i \sin \frac{\pi}{3}\right)$
• D. $4\left(\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}\right)$

Answer 1

Answer: (C)

Let $1=r \cos \theta, \sqrt{3}=r \sin \theta$
By squaring and adding, we get
$r^{2}\left(\cos ^{2} \theta+\sin ^{2} \theta\right)=4$
i.e, $r=\sqrt{4}=2$
Therefore, $\cos \theta=\frac{1}{2}, \sin \theta=\frac{\sqrt{3}}{2}$,
which gives $\theta=\frac{\pi}{3}$
Therefore, required polar form is :
$z=2\left(\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}\right)$