If you look at the Wikipedia article on big-O notation, it says that f(x) = O(g(x)) if and only if there exists some M > 0 and x_0 such that |f(x)| < M|g(x)| for all x > x_0. The x_0 is what lets you ignore early irregularities; without it, you couldn't say that sin(x) + 1 is O(x) since there's obviously no M such that sin(0) + 1 < M * 0.

You're right about the theta thing, my mistake.