 aggierabz2006zw

2022-07-14

Strategies to work with system of trigonometric inequality
$\left[\mathrm{exp}\left(-{q}_{1}\ast i\right){\mathrm{cos}}^{3}\left({p}_{3}\right)\mathrm{sin}\left({p}_{1}\right)\mathrm{sin}\left({p}_{2}\right)\mathrm{sin}\left({p}_{3}\right)\right]\cdot a-\left[\mathrm{exp}\left(-{q}_{2}i\right)\mathrm{exp}\left(-{q}_{3}i\right)\mathrm{cos}\left({p}_{1}\right)\mathrm{cos}\left({p}_{2}\right){\mathrm{cos}}^{2}\left({p}_{3}\right)\mathrm{sin}\left({p}_{2}\right){\mathrm{sin}}^{2}\left({p}_{3}\right)\right]\cdot b\ne 0,$
where $a$ and $b$ are complex variables and ${q}_{i}$ and ${p}_{i}$ are real variables. The real system (the least of them) have 18 inequalities and 8 variables, I need know if there is a set of values (8 real values) that makes true at least one of the inequalities. Maybe a good path is know how determine the minimum value of a expression of type
$\left[\mathrm{exp}\left({a}_{1}i\right){\mathrm{cos}}^{3}\left({a}_{2}\right)\mathrm{sin}\left({a}_{3}\right)\mathrm{sin}\left({a}_{4}\right){\mathrm{sin}}^{2}\left({a}_{5}\right)\right]$
or a bit more complex like
$\left[\mathrm{exp}\left({a}_{1}i\right){\mathrm{cos}}^{3}\left({a}_{2}\right)\mathrm{sin}\left({a}_{3}\right)\mathrm{sin}\left({a}_{4}\right){\mathrm{sin}}^{2}\left({a}_{5}\right)\right]\cdot \left[\mathrm{exp}\left(-{b}_{1}i\right){\mathrm{cos}}^{2}\left({b}_{2}\right)\mathrm{sin}\left({b}_{3}\right)\mathrm{sin}\left({b}_{4}\right){\mathrm{sin}}^{3}\left({b}_{5}\right)\right].$
Then, some idea? Jamarcus Shields

Expert

If you take $a=b$, and ${p}_{i}={q}_{i}=r$, your inequality reduces to
$0\ne a{e}^{-2ir}{\mathrm{cos}}^{3}r{\mathrm{sin}}^{3}r\left({e}^{ir}-\mathrm{cos}r\right)=ai{e}^{-2ir}{\mathrm{cos}}^{3}r{\mathrm{sin}}^{4}r$
This is satisfied by any $a\ne 0$ and $r\ne n\pi /2$ (for integer $n$). But that's really over-thinking things. Agostarawz

Expert