Possible statements to prove that $x^2 + y^2 =1$ is parametrized by $f(t) = \langle \cos t, \sin t \rangle$

Question

I'm trying to find a meaningful mathematical statement to prove that would correspond to what we intuitively mean when we say "the graph of $x^2 + y^2 =1 $ is identical to the graph of $f(t) = \langle \cos t, \sin t \rangle$", or "any point in the former's graph is also a point in the latter's graph". I'm wondering if establishing the following statement would indeed tell us that there is a perfect correspondence between the two: $\{ x,y: \forall t \in \mathbb{R}, x = \cos t \land y = \sin t \}$ is a solution for $x^2 + y^2 = 1$.

Is this properly stated? Would there be another way to state it more exactly to achieve the aim of establishing an "identity" between $x^2 + y^2 = 1$ and $f(t) = \langle \cos t, \sin t \rangle$, e.g. do we usually mean something else by "is parametrized by"? Any tips and help on this question are highly appreciated!

Answer 1

One way would be$$\{(x,y)\in\Bbb R^2\mid x^2+y^2=1\}=\left\{\bigl(\cos(\theta),\sin(\theta)\bigr)\,\middle|\,\theta\in\Bbb R\right\}.$$

What you wrote is not correct. For instance, what does it mean to say that a set a solution of an equation? However, it would be correct to say that it is the set of all solutions.

Answer 2

In many courses, the circle functions are defined by this very relation. For any $t$ between $0$ and $2\pi$, $\cos t$ is the $x$-coordinate of the point on the unit circle measuring $t$ radians counterclockwise from $(1,0)$, and $\sin t$ is the $y$-coordinate of the same point. In this situation, the only justification I would look for is “By definition.”