I know that the locus of $\mathrm{arg}(z)=\theta$ is a half line with angle $\theta$, but I'm not sure why?
I can start the proof: $ z=x+iy $ $ \theta=\mathrm{arg}(z)=\arctan\left(\frac{y}{x}\right) $ $ \tan(\theta)=\frac{y}{x} $ $ y=x\cdot \tan(\theta) $ Which tells me that the locus is a line with gradient $\tan(\theta)$ passing through $(0,0)$, but I know that it should be a half line with gradient $\tan(\theta)$ starting at $(0,0)$.
Why is this?