I have recently begun studying Numerical Analysis and have been given the following problem:
For small values of $x$, how good is the approximation $\cos(x)\approx 1$? How small must $x$ be to have $\frac{1}{2}\cdot 10^{-8}$ accuracy?
I am unsure what kind of answer is expected for the first question. For the second question I assume that I have to use the Taylor series for $\cos(x)$. Beyond that, I really don't have a clue. Can anybody guide me through it?
By taylor's theorem, it holds that $\cos(x) = 1 + R_1(x)$. In the Lagrange's form of the remainder, there exists $c$ such that $R_1(x) = \frac {(-\cos(c))} {2!} \cdot x^2$.
$|R_1(x)| = |\frac {(-\cos(c))} {2!} \cdot x^2| = \frac {|\cos(c)|} 2 x^2 \leq \frac 1 2 x^2$
so $|R_1(x)| \leq \frac 1 210^{-8} \iff \frac 1 2 x^2 \leq \frac 1 2 10^{-8} \iff x^2 \leq 10^{-8} \iff -10^{-4} \leq x \leq 10^{-4}$