Solve nonlinear system of equations Solve the system of equations $$\begin{cases}163-400z\sin{x}&=0\\-135z+85\cos{x}+61&=0\end{cases}$$
What is the best way of going about this?
I rearranged the second equation for $z$ and then substituted it into the first one to find $$163-400\left(\frac{1}{135}(85\cos{x}+61)\right)\sin{x}=0$$ $$\implies 22005 - 34000\sin{x}\cos{x}-24400\sin{x}=0$$ $$\implies 4401-6800\sin{x}\cos{x}-4880\sin{x}=0$$ $$\implies 80\sin{x}(85\cos{x}-61)=4401$$ Now how do I solve this?
 A: HINT: use that $$\sin(x)=2\,{\frac {\tan \left( x/2 \right) }{1+ \left( \tan \left( x/2
 \right)  \right) ^{2}}}
$$ and
$$\cos(x)={\frac {1- \left( \tan \left( x/2 \right)  \right) ^{2}}{1+ \left( 
\tan \left( x/2 \right)  \right) ^{2}}}
$$
A: Robert Israel gave the rigorous solution of the problem solving the quartic equation in $s$.
Outside this approach, only numerical methods would lead to the solution; probably, Newton method could be the simplest.
Looking at the plot of function $$f(s)=46240000 s^4-22425600 s^2-42953760 s+19368801$$ $(-1 \leq s \leq 1)$, you can notice that there is one root close to $0.4$ and another close to $1.0$.
So, let us use Newton method with these starting guesses.
For the first root, the iterates will be $$x_1=0.395575435005952$$ $$x_2=0.395584036761436$$ $$x_3=0.395584036792970$$ which is the solution for fifteen significant figures.
For the second root, the iterates will be $$x_1=0.997638403524923$$ $$x_2=0.997623606632696$$ $$x_3=0.997623606053749$$ which is the solution for fifteen significant figures.
