Is the circle a vector-valued function?

Question

I know the circle is not representable by a function $f: \mathbf{R} \to \mathbf{R}$.

However, if I take the vector function $F(t) = (\sin{t}, \cos{t})$, can that be considered the function of a circle?

Answer 1

Yes, it can. Sometimes we call your function $F\colon\mathbb{R}\rightarrow\mathbb{R}^2$ given by $$F(t) = (\cos t, \sin t)$$ a parametrisation of the circle. It is a vector-valued function, whose image is the circle.