I want to this vector values function to an XY Cartesian function
$$\bf V(t)= (\cos(t)+t\sin(t), \sin(t)-t \cos(t))$$
I know the answer for
$$\bf V(t)= (\cos(t),\sin(t))$$
it is
$$y(x) = \pm \sqrt{1-x^2}$$
I cannot see how to generalise to the one above?
$V(t)$ is a spiral that looks Archimedean on first sight but is not. The distance between successive turns is a variable that approaches $\pi$.
Expressing the spiral in Cartesian form $y=f(x)$ is impossible, both because one x value can correspond to multiple y values (the relation is not obvious either) and because of the $t\sin t$ and $t\cos t$ terms. However, if you treat the output of $V(t)$ as a complex number then $$V(t)=(1-it)e^{it}$$ and this is the form I would prefer.