\(z = \frac{1 + \sqrt3 i}{1 +i} \)
\(= \frac{1 + \sqrt3 i}{1 +i} \times \frac{1-i}{1-i} \)
\(= \frac{(1 + \sqrt3) + i(\sqrt 3 - 1)}{1 -i^2}\)
\(= \frac{1 + \sqrt 3}2 + i \frac{(\sqrt 3 - 1)}2\)
\(|z| = \left(\frac{\sqrt 3 + 1}2\right) ^2 + \left(\frac{\sqrt 3 - 1}2\right) ^2 \)
\(= \frac{(3 +1 + 2\sqrt3) + (3 + 1- 2\sqrt3)}{4}\)
\(= \frac 84 = 2\)
\(tan\theta =\frac ba = \cfrac{\frac{\sqrt3 - 1}2}{\frac{\sqrt3+1}2}\)
\(= \frac{\sqrt3 - 1}{\sqrt3+1}\)
\(= \frac{\sqrt3 - 1}{1 + \sqrt3 \times 1}\)
\(= \frac{tan60° - tan45°}{1 + tan60°tan45°}\) \((\because tan60° = \sqrt 3)\)
\(= tan(60° - 45° )\)
\(= tan(15° )\)
\(\therefore \theta = Arg\, z = 15°\)
