From the solution of the equation:
Solve[a Sin[\[Theta]/2]^2 + b Sin[\[Theta]/2] Cos[\[Theta]/2] + c == 0, \[Theta]] // Simplify // PowerExpand
Take the inequality:
$\sqrt{b^4-4b^2c(a+c)}>0$
The solution for it is https://www.wolframalpha.com/input?i=sqrt%28b%5E4-4b%5E2c*%28a%2Bc%29%29%3E0+solve+for+a%2Cb%2Cc%2CReals:
$\frac{1}{2}(-\sqrt{a^2+b^2}-a)<c<\frac{1}{2}(\sqrt{a^2+b^2}-a)$
$a, b$ and $c$ are defined as follows:
$a=2 \cdot (M \cdot M \cdot x) \cdot y$
$b=2 \cdot(M \cdot x) \cdot y$
$c=x \cdot y$
here $x,y$ are arbitrary unit vectors
$M$ - skew-symmetric matrix
My intuition tells me that one can move from mathematical abstraction to a visual geometric representation of the solution of this inequality. Either some kind of vector diagram is spinning in my head, or something with a parallelepiped or a circle.
I need a hint: how can I display the solution of the inequality in this case?
EDIT:
I discovered something.If we build two-dimensional graphs (with a fixed $a$), then we can see the following.
RegionPlot[{Evaluate@Table[b^4 - 4 b^2 c (a + c) > 0, {a, 1/2, 1, 0.25}]}, {b, -2, 2}, {c, -2, 2}, ImageSize -> Small, PlotPoints -> 50]
It can be seen that the region for solving the inequality is an "hourglass" with a certain width of the "neck", which is $=a$. Thus, if the parameter $c$ is within this "hourglass", the inequality is satisfied. It should be noted that the smaller $a$, the smaller the width of the neck and the higher the risk that the inequality may not be feasible, since the risk of $c$ falling out of the hourglass is higher.