Show that $x(t)=2r\cos^2(t)$, $y(t)=2r\sin t\cos t$ is a regular parametrization of the real circle of radius $r$, centre $(r,0)$.

Question

Show that $x(t)=2r\cos^2(t)$ and $y(t)=2r\sin t\cos t$ is a regular parametrization of the real circle of radius $r$, centre $(r,0)$.

I set up the equation of the circle as follows. $(x-r)^2+y^2=r^2$

And I set the coordinates of the circle as follows.

$x(t)=r+\cos t$, $y(t)=r\sin t$

What should I do to solve this problem from now on?

Answer 1

Because of the identities $\cos^2 u = (1 + \cos 2u)/2$ and $\sin 2u = 2\sin u \cos u$, your parametric equations are equivalent to

$$\begin{align}x(t)&=r + r\cos 2t\\y(t)&=r \sin 2t\end{align}$$

And now you can see that this describes the circle centred $(r, 0)$ with radius $r$.

Yes, the parameterisation is twice as fast as you'd expect (with $2t$ in the expressions), but this is okay as parametrisations are not unique.