\text{Arg}[z]=\text{ArcTan}\left[\frac{\text{Im}[z]}{\text{Re}[z]}\right]=\text{ArcTan}\left[\frac{\text{Sin}\left[\frac{x}{3}\right]- \text{Sin}\left[\frac{x}{6}\right]}{\text{Cos}\left[\frac{x}{3}\right]+\text{Cos}\left[\frac{x}{6}\right]}\right]=\text{ArcTan}\left[\text{Tan}\left[\frac{x}{12}\right]\right]=\frac{x}{12}