(i) Define in polar coordinates $r = f(\alpha)$ the origin-centred circle with radius $R$. Specify the domain range for the polar coordinate $\alpha$.
(ii) Define in polar coordinates $r = f(\alpha)$ a circle with radius $R$ and the centre at the Cartesian coordinates $(R, 0)$. Specify the domain range for the polar coordinate $\alpha$.
My Answer for (i):
$x = r \cdot \cos \alpha$
$y = r \cdot \sin \alpha$
$x^2 + y^2 = R^2$ (equation for circle, radius R centered at origin)
$r^2(\cos^2 \alpha + \sin^2 \alpha) = R^2 $
$r^2 = R^2$
$r = R$
The domain range of $\alpha$ is $(-\infty, \infty) $?
Is my answer correct? I don't think it's correct because my RHS doesn't have $\alpha$ at all. I've learnt that domain is the set valid input values and range is the set valid output values. However, I don't understand what the question means by "domain range" of $\alpha$?
For part (ii) I don't understand how do you shift a circle centered at origin to become the radius.
I am sorry if this seems like a trivial question, please don't close this question. I haven't done any serious math for the last 2 years, especially geometry. The textbook that he had written (and made us buy) didn't really ease us into the topic but expected us to already know all these formulas, I only managed to do these much after 6 hours of googling and reading up on functions, polar equations, and so on. Please please please help, thanks!!