The small-angle approximation for cosine is: $$ \cos (x) = 1 - \frac{x^2}{2} $$ Question: How can I find a range of values of $x$ for which this approximation gives correct results rounded to 2 decimal places?
Thought: The error term of this $2^{nd}$-order Taylor approximation is $$ E(x)=\frac{sin(\eta)}{6}x^3, $$ where $\eta$ is between $x$ and $0$. Thus, $$ |E(x)|<10^{-2} \to |\sin(\eta)x^3|<6\times10^{-2}. $$ This is just my thought, but I am not sure this is the correct approach.
The alternating series theorem says the truncation error is smaller than the first neglected term and of the same sign. The first term you neglect is $\frac {x^4}{4!}$ so we want $$\frac {x^4}{4!} \lt 0.01\\x^4 \lt 0.24\\|x|\lt 0.24^{1/4}\approx 0.700$$ When you demand correct rounding to a number of places it is hard to say what the allowable error is. If you are very close to a breakpoint you may have to be very accurate. I used $0.01$ as the allowable error, you can use whatever value you want.