Let $f: \mathbb{R} \rightarrow \mathbb{S}^{1} \times \mathbb{S}^{1}$ given by:

$f(t)=(\exp(it),\exp(qit))$.

I want to show that $f$ is an immersion. OK I know the definition: we compute its derivative and check it is injective.

My question is: can we view the map as $f(t)=(\cos(t),\sin(t),\cos(qt),\sin(qt))$ i.e $f: \mathbb{R} \rightarrow \mathbb{R}^{4}$ or do we have to introduce charts? I'm familiar with maps $g: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ but here we have the torus and the map in terms of complex numbers so I'm confused.

Can you please explain how to see $f$ is an immersion? What are the steps?