# Represents the complex number $\;z=1+ \sqrt{3} i \;$ in the polar form

$(a)\;2(cos \large\frac{\pi}{3}+ i sin\large\frac{\pi}{3})\qquad(b)\;2(cos \large\frac{\pi}{5}+ i sin\large\frac{\pi}{5})\qquad(c)\;(cos \large\frac{\pi}{3}+ i sin\large\frac{\pi}{3})\qquad(d)\;0$

Answer : $\;2(cos \large\frac{\pi}{3}+ i sin\large\frac{\pi}{3})$
Explanation :
Let $\; r cos \theta = 1 \;$ and $\; r sin \theta = \sqrt{3}$
By squaring and adding , we get
$r^{2} (cos^{2} \theta + sin^{2} \theta)^{2} = 4$
$r = \sqrt{4} = 2 \qquad$ [conventionally , r > 0]
Therefore , $\; cos \theta = \large\frac{1}{2}\; , sin \theta = \large\frac{\sqrt{3}}{2}\;,$ which gives $\; \theta = \large\frac{\pi}{3}$
Therefore , required polar form is $\;2(cos \large\frac{\pi}{3}+ i sin\large\frac{\pi}{3})$
the complex number $\;z=1+ \sqrt{3} i \;$ is represented as shown in figure .