Prove that the set of polynomials is a vector space
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Your map is periodic of period $2\pi$ in both variables, so you can factor it through the quotient $\mathbb R^2\to\mathbb R^2/\mathbb Z^2$ to obtain a map $\phi:\mathbb R^2/\mathbb Z^2\to\mathbb R^5$ with exactly the same image.
Now this map $\phi$ is an immersion (because, as you observed, the original map has differential with full rank) and it is not hard to see that it is injective. Since its domain is compact, the image is a submanifold.
(Of course, one has to check this last claim!)
What you need is the inverse function theorem: http://en.wikipedia.org/wiki/Inverse_function_theorem
To satisfy the conditions of the theorem, you just need to check that the total derivative of your map never has rank equal to less than 2, which follows from the fact that $\det D\phi^T D\phi$ is always nonzero.