aggierabz2006zw

Answered

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?

Answer & Explanation

Jamarcus Shields

Expert

2022-07-15Added 17 answers

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

2022-07-16Added 3 answers

Being equal to zero is hard, so almost any values of the variables will make the expression nonzero.

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get your answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?