I am trying to parametrize the following spiral in polar coordinates. My book says
$$ \gamma = (t\cos t, t\sin t) ,\quad t \in [0,+\infty]$$
can be parametrized in polar coordinates as $$\gamma_p = (t, t) , \quad t \in [0,+\infty]$$
But I am getting nowhere trying to prove that. I am trying to do this as a way to understand the following:
A curve in the plane can be defined by a continuous function
$$\gamma_p: I \subseteq \mathbb R \to [0,\infty] \times \mathbb R$$ where $$\gamma_p= (r(t), \theta(t))$$
are the polar coordinates of a point in the image of the curve associated to the parameter $t$, which corrisponds to the curve
$$\gamma= (r(t)\cos(\theta(t)), r(t)\sin(\theta(t)))$$ in cartesian coordinates.
which I am having trouble understanding because they are mixing polar coordinates, cartesian coordinates and parametrization and I don't see how polar coordinates can be seen as a curve
Idea is: given the point $(x, y) = (t \cos t, t \sin t)$ how do you represent it in polar coordinates? You just need to calculate the values for $(r, \theta)$, where
$$ r = \sqrt{x^2 + y^2} = \sqrt{(t\cos t)^2 + (t\sin t)^2} = t \\ \tan \theta = \frac{y}{x} = \frac{t \cos t}{t \sin t} = \tan t $$
So you have that the representation of $(x, y) = (t \cos t, t \sin t)$ in polar coordinates is $(r, \theta) = (t, t)$. The spiral is just all the collection of all these points for $t \ge 0$