Given a system of linear equations with $N$ unknowns an $M$ equations $(N<M)$ its not too difficult to see if there is a solution.
Now consider an extended system of equations where a term in an equation can be either a simple variable or $\sin()$ or $\cos()$ of a variable (or its multiple). The power of each term is still 1. For eg:
$15\sin(10x)-5y+3\cos(7z)=0$
Given any problem instance, can this system be solved systematically/easily? The number of unknowns is still greater than the number of equations.
Edit: My background is CS and not mathematics and this is a problem that arose out of a self-study scenario (from pure Linear algebra). So not much to go about in external help. I am unaware what this set of equations are actually called so couldn't look for help online either.
I have a little matured my thoughts : IMHO, nothing can be said in general, no general methodology can be applied.
Let us consider for example the system you have given:
$$\begin{cases}15\sin(10x)-5y+3\cos(7z)&=&0\\ 23\sin(7x)-17y+5\cos(19z)&=&0\\ 24\sin(6x)-17\cos(37y)-35\cos(16z)&=&0\end{cases}$$
Let us use a classical "trick" : fix a value of $z$ (which is parameter $a$ for Desmos), for understanding the kind of complexity we have to deal with :
You can see how strange some of the curves (the black one here) may behave, for a given value of $a$ (here $a=1.5$). When $a$ moves, one can spot values where, apparently, the three curves have (at least) a common point, but how complicated it is with only 3 variables !
The complexity is enhanced by the fact that a variable such as $y$ can appear "naked" (equations (1) and (2)) or "coated" with a trigonometric function, here $\cos(37y)$ in equation (3). I think you have to avoid this mixed use.