<s>[INST] Searchingthe polar form of $z = 2+i(1+\sqrt{3})$ Find the polar form of:
$$z = 2+i(1+\sqrt{3})$$
Is there any other way to solve that in different way than by finding $|z|$ that is really "ugly" ($|z| = \sqrt{8+2\sqrt{3}}$)?
 [/INST] you are right  $z=\rho \cdot e^{i\cdot \theta}$ with  $\rho=|z| = \sqrt{8+2\sqrt{3}}$. But Note that  $$ \frac{1 + \sqrt3}{2} = \cos\frac{π}{3}+\sin\frac{π}{3}=\sqrt2\cos(\frac{π}{3}-\frac{π}{4} )=\sqrt2\cos\frac{π}{12}$$
and then , 
$$\theta=\arg (z)=\arctan\left(\frac{1 + \sqrt3}{2}\right)=\color{red}{\arctan\left(\sqrt2\cos\frac{π}{12}\right)}$$

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