I have the following trigonometric equation $f(\theta)=100(A_2 B_3 - A_3 B_2)^2 - (c_1B_3 - c_2 B_2)^2 - (c_2A_2 - c_1 A_3)^2=0,$
where:
$ A_2 = 3\cos(\theta)-5$
$B_2 = 3\sin(\theta)$
$A_3 = 3(\cos(\theta) - \sin(\theta))$
$B_3 = 3(\cos(\theta) + \sin(\theta))-6$
$c_1 = p_2^2 - 25 - A_2^2 - B_2^2$
$c_2 = -16 - A_3^2 - B_3^2$
and need to find all the values of $ p_2 $ for which $f(\theta)=0$ has 2, 4, 6 solutions in $[-\pi,\pi]$ and no solution. For example, if $ p_2 = 4 $, $f(\theta)=0$ has 2 solutions in $[-\pi,\pi]$. If $ p_2 = 5$, $f(\theta)=0$ has 4 solutions. If $ p_2 = 7 $, $f(\theta)=0$ has 6 solutions. If $ p_2 = 1$, $f(\theta)=0$ has no solution.These values follow from the graph of the function $f(\theta)$ (tested on MATLAB). So what if we want to generalize this approach? Any ideas?