I might be missing some background knowledge on this subject, but nevertheless I am interested. In some cases like this, the answers talk about finding the taylor series for $\cos(x)$ and then substituting the first two terms of the series in instead of $\cos(x)$ i.e in the link they solve for $x$ in
$$\frac{x^2}{2} \leq \frac{1}{2}\cdot 10^{-8}$$
Rather than $$1-\cos(x) \leq \frac{1}{2}\cdot 10^{-8}$$
So my question is - why is this allowed? And why not include the 3rd term from the Taylor series as well?
This is allowed because the Taylor series for $\cos x$ is an alternating series, and they use Leibniz' theorem for alternating series: the error is bounded by the first missing term (in absolute value) and has the same sign.