I am trying to calculate **analytic** solution (or *locus*) of *zeros* of a very large **multi-variable** function which is consisted of thousands of **nonlinear trigonometric** terms. All the variables are **real numbers**. The function **is not differential**.


(The equation is a *Singular Value* of the *constrains matrix* of a mechanical system)


Here is an example of what I want:




---


For this equation:
$$
 f = y\sin(\theta) - z\cos(\theta)
$$
Where $ \theta,y,z \in \mathbb{R} $ (and if it helps: $ 0 \le \theta < 2\pi$).


I want all the sets of **real** values of $\theta$, $y$ and $z$ that satisfy $f = 0$, which would be the following three set of answers:


$$ \theta = \arctan(\frac{z}{y}) $$


$$ z = 0,\ \theta = \{0,...\},\ y\in \mathbb{R} $$


$$ y = 0,\ \theta = \{\pi/2,...\},\ z\in \mathbb{R} $$


**Q**: Is there any way in any software that this solution could be achieved?


Any help would be much appreciated! Thanks.



Finding zeros of a multi-variable nonlinear trigonometric function
In the particular case you give, you can make the substitution $t = tan(\theta)/2,$ which transforms your equation into a single polynomial in three variables, of which you want to find all *real* solutions (you can then back out the $\theta$). If your trig functions depend nicely on the argument, this reduction will always work, at which point (since you say "function") you want to find all real real zeros of a multivariate polynomial. I am not sure what form you want for these solutions (generally there will not be a nice parametrization), but Mathematica's Solve[] or Reduce[] will often give you something useful.


