Strategies to work with system of trigonometric inequality [ exp ⁡ ( − q 1...

aggierabz2006zw

aggierabz2006zw

Answered

2022-07-14

Strategies to work with system of trigonometric inequality
[ exp ( q 1 i ) cos 3 ( p 3 ) sin ( p 1 ) sin ( p 2 ) sin ( p 3 ) ] a [ exp ( q 2 i ) exp ( q 3 i ) cos ( p 1 ) cos ( p 2 ) cos 2 ( p 3 ) sin ( p 2 ) sin 2 ( p 3 ) ] b 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
[ exp ( a 1 i ) cos 3 ( a 2 ) sin ( a 3 ) sin ( a 4 ) sin 2 ( a 5 ) ]
or a bit more complex like
[ exp ( a 1 i ) cos 3 ( a 2 ) sin ( a 3 ) sin ( a 4 ) sin 2 ( a 5 ) ] [ exp ( b 1 i ) cos 2 ( b 2 ) sin ( b 3 ) sin ( b 4 ) sin 3 ( b 5 ) ] .
Then, some idea?

Answer & Explanation

Jamarcus Shields

Jamarcus Shields

Expert

2022-07-15Added 17 answers

If you take a = b, and p i = q i = r, your inequality reduces to
0 a e 2 i r cos 3 r sin 3 r ( e i r cos r ) = a i e 2 i r cos 3 r sin 4 r
This is satisfied by any a 0 and r n π / 2 (for integer n). But that's really over-thinking things.
Agostarawz

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.

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