calculus - How to find the $arctan(2sqrt{3})$ by hand?

I'm trying to find the polar form of the complex number $zw$ where $z = 1 + i$. and $w = \sqrt{3} + i$.

I multiplied foiled the complex numbers, grouped the real and imaginary terms together to get a modulus of $\sqrt{8}$ and an angle of $\theta = \arctan(2\sqrt{3})$. I dont know how to find this, i do know that $\arctan(\sqrt{3})$ is $\pi/3$ but i dont know how to incorporate the multiplied 2. The answer is given as $5\pi/12$.