Why $g(t)=(\cos t, \sin t)$ and $h(t) (\cos 2t, \sin 2t)$ have the same image?

Question

I know this might sound silly, but I don't know why $g(t)=(\cos t, \sin t)$ and $h(t)= (\cos 2t, \sin 2t)$ have the same image, that is the unit circle. I understand that if we eliminate the parameter on $g$, we get $x=\cos t$ and $y=\sin t$, thus $x^2+y^2=1$ which is the equation of the unit circle, but why h has the same image if it is a different function?

Answer 1

Let me illustrate why on the simple functions $f(x)=x$ and $g(x)=2x$ defined on $\mathbb R$.

These two are obviously different functions but they both have $\mathbb R$ as their image. This means that for every real number $y$ there exists a real number $x$ such that $f(x)=y$, and similarly for $g$.

Take some $y\in \mathbb R$. The function $f(x)$ will have that value when $x=y$. The function $g(x)$ will have that value when $x=y/2$. But, for any value you pick, both functions will take that value eventually, just for different values of their input variables.

Now you can apply this reasoning to your problem.