Solving a system of equations/inequalities that have trigonometric functions on the left-hand side - MathOverflow

Solving a system of equations/inequalities that have trigonometric functions on the left-hand side

2012-01-27T17:56:41Z

Is there any known (symbolic) method that solves a system of equations/inequalities that have trigonometric functions on the left-hand side of the system?

Ex) Find $x,y,\theta \in \mathbb{R}$ that satisfy

\begin{align} 2x + y + 3\cos(\theta) - 2\sin(\theta) \le& 0 \\ x - y + 4\cos(\theta) + 2\sin(\theta) \le& 0 \end{align}

Answer by SJR for Solving a system of equations/inequalities that have trigonometric functions on the left-hand side

2012-01-27T18:47:22Z

Using, e.g., the sin function, one can write a system of inequalities in a given variable $x$ that is satisfied if and only if $x$ is an integer. Therefore, an algorithm for solving inequalities of the kind you asked about would give an algorithm of finding all integer solutions to an arbitrary system of inequalities. This would contradict the negative solution to Hilbert's Tenth Problem. See http://en.wikipedia.org/wiki/Hilbert%27s_tenth_problem

Answer by zeb for Solving a system of equations/inequalities that have trigonometric functions on the left-hand side

2012-01-27T23:45:54Z

Here's some good news to balance out the bad: If every variable that appears inside a $\sin$ or a $\cos$ is bounded, then there is something very close to a method for deciding such problems.

A basic result is that the theory of the real numbers with addition, multiplication, and the function

$S(x) = \cases{0, |x|>1\\ \sin(x), |x|\le 1}$

is strongly model-complete. This was proved by Lou van den Dries, in a paper titled "On the Elementary Theory of Restricted Elementary Functions" (link: http://www.jstor.org/stable/2274572).

There is also an actual algorithm due to Adam Strzebonski for deciding such problems, but its correctness depends on Schanuel's conjecture, which is currently open. (link: http://dl.acm.org/citation.cfm?id=1576749.)