I know that Taylor's theorem is a decent approach to this problem, but I'm looking for assistance in setting it up. My function is $f(x)=\mbox{sin}(x)$, but what should my initial guess be? $49^\circ$, maybe? And is there a way to predetermine the degree of the Taylor approximation needed to be within the requested tolerance?
2026-04-22 09:42:01.1776850921
Derive an expression $E$ for an approximation of $L=$ sin($50^{\circ}$) satisfying $|E-L|<0.0001$
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Your initial guess should be a nearby point where it is easy to compute the function and its derivatives. $45^\circ$ is a good candidate. You probably know the value and derivatives of $\sin x$ at that point. For the number of terms, see the error term in Taylor's theorem or the alternating series theorem.