Convert to parametric equation from implicit equation? Given an implicit equation such as $x^2+y^2=1$ , I know it corresponds to the parametric equations
$
\begin{cases}
x=\cos t\\
y=\sin t
\end{cases}$.
But I don't know how to get from the implicit equations to the parametric one, only the other way around.
I'm interested in a general algorithm in solving for the parametric equations given the implicit equation. If such algorithm doesn't exist I want to know how to approach this problem given different types of implicit equations:

*

*Polynomial

*Trigonometric

*Exponential

*Other types and/or a combination of any of the above

Maybe some of them can be transformed and some others can't (How can I tell?).
To illustrate such an algorithm, please show me how to get parametric equations for $y\sin x+x\cos y=1$.
Thank you!
 A: The parametric equation $\begin{cases}x=\cos t\\ y=\sin t\end{cases}$ you wrote, is actually the entire solution set of the implicit equation $x^2+y^2=1$.  In this sense, if we cannot solve the implicit equation in closed-form using standard mathematical functions with respect to any variable, then we cannot obtain the parametric equation either.
Let's consider the variable $y$, as a fixed number. We want to see the solution of the equation $y\sin x+x\cos y=1$, with respect to $x:$
$$\begin{align}&\sin x=\frac {1-x\cos y}{y}\\
\implies &\sin x=-\frac {\cos y}{y}x+\frac 1y\end{align}$$
Then, note that the equation $\sin x=kx,~k≠0$ (except $x=0$) and $\sin x=ax+b$, where $a,b≠0$ are types of transcendental equations that do not have a general closed-form solution.
Therefore, the implicit equation $y\sin x+x\cos y=1$ cannot be expressed by means of closed - form parametric equations, in general.

However, the equations we noted as transcendental, do not have only general closed-form solutions. For instance, the specific transcendental equation $\sin x=\frac 3\pi x$ have closed-form solutions. Indeed, $x_1=0,~x_2=\frac \pi 6,~x_3=-\frac \pi 6$ are only possible solutions.
To summarize, the implicit equation $y\sin x+x\cos y=1$, cannot be expressed with parametric equations, even through any special functions defined in mathematics.
