The polar form of any complex number can be written as
[tex]z = |z| e^{i\arg(z)}[/tex]
where [tex]\arg(z)[/tex] is the argument of [tex]z[/tex], i.e. the angle it makes with the positive real axis in the complex plane.
If [tex]z=\sqrt3+i[/tex], then [tex]z[/tex] has modulus
[tex]|z| = \sqrt{\left(\sqrt3\right)^2 + 1^2} = \sqrt4 = 2[/tex]
and argument
[tex]\arg(z) = \tan^{-1}\left(\dfrac1{\sqrt3}\right) = \dfrac\pi6[/tex]
Then
[tex]\sqrt3 + i = 2e^{i\frac\pi6} = 2 \left(\cos\left(\dfrac\pi6\right) + i \sin\left(\dfrac\pi6\right)\right)[/tex]
