How to make the angular parameterization of $x^2+y^2=1$ unique?

Question

If there exists a relation $x^2+y^2=1$ between two real variables $x$ and $y$, one can always make the parameterization $x=\pm\cos\theta$ and $y=\pm\sin\theta$. Can we make the parameterization unique i.e., $x=\cos\theta$ and $y=\sin\theta$ by choosing $\theta$?

Answer 1

Choosing the interval $0\le \theta <2\pi$ the parametrization is $(x,y)=(\cos \theta, \sin \theta)$ and gives the entire circle.

Answer 2

Yes, but you need to define a function which compilers (and Mathematica and Excel) call "Atan2" which is the arctangent of the fraction represented by its two arguments (in Mathematica and C and C++ they would be $y$ then $x$, in Excel they would be $x$ then $y$, so watch out), in the quadrant determined by the signs of the two arguments.

So for example, Atan2 ( -1, -1 ) is $5\pi/4$ rather than $\pi/4$.

With this function, the $\theta$ you want is merely Atan2($y,x)$.

BTW, Atan2 always returns an answer in $-\pi/2 < \theta \leq \pi/2$.