Calculate $\sin(0.2)$ to 4 decimal places with its Taylor series and the Lagrange error bound

Question

So, I have to estimate the value of sin(0,2) with at least 4 correct decimal places by using its Taylor series and the Lagrange error bound, but I am stuck in what I believe is a trivial step.

I understand the following:

$$\sin(x)=\sum_{n=0}^∞(-1)^n\frac{x^{2n+1}}{(2n+1)!}$$

$$f(x)=P_{n}(x) + R_{n}(x)$$ so, $$\sin(0,2)=P_{n}(0,2)+R_{n}(0,2)$$

By having that, I assume that I should find the value for n so that $$f^{(n+1}(x)\leq M$$ where M (I guess?) is 1, the maximum value for $\sin(x)$

Knowing that, My Lagrange error should be:

$$R_n(x)\leq\frac{M(x)^{n+1}}{(n+1)!}$$ so: $$R_n(0,2)\leq\frac{(0,2)^{n+1}}{(n+1)!}\leq 0,0001$$ By checking values for n, I found that for n=3, its value is 0,00006..., and so I chose 3 as the degree of the Taylor polinomial that should give me an error smaller than 0,0001

Now I am stuck at calculating sin(0,2), which I assume it should be like this, where I do not know how to express Rn(x), nor if the n I chose is correct, since: $$\sin(0,2)=\frac{(0,2)^{2n+1}}{(2n+1)!}=\frac{(0,2)^{(2*3)+1}}{((2*3)+1)!}=2.539682539682542*10^{-9}$$ which is nowhere near its "correct" value, which I found that it's best approximated by making n=0: $$\sin(0,2)=\frac{(0,2)^{2n+1}}{(2n+1)!}=\frac{(0,2)}{1!}=0,2$$

If I take n=0, then I do not know how to even start writing Rn(x): $$\sin(0,2)=\frac{(0,2)^{2n+1}}{(2n+1)!}=\frac{(0,2)}{1!}=>0,2-R_n(x)=0,2-what?$$

Any help and corrections are greatly appreciated, I am also unsure if I chose the value for n correctly.

Answer 1

It looks like you're using the 3rd-order term, but you need to use the whole series up to the 3rd-order term. Since the 0th- and 2nd- order terms are $0$, you just need to add the 1st-order term, which is $x$.

Second, to get 4 correct decimal places, the error really needs to be less than $0.00005.$

Third, if you're using $2n+1$ as the index in your series, be sure to use $2n+1$ in the remainder term, rather than just $n$.

Answer 2

Since you found that $n=3$ is the degree of the Taylor polynomial that satisfies the error bound, your approximation should be $$\sin(0.2)=0.2-\frac{0.2^3}{3!},$$ in which you use $P_3(x)$ to approximate $\sin(x)$, based on your notation.

Answer 3

$$\sin x=x-\frac{x^3}{3!}+R_n$$ You calculated that for $n=3$ $R_n<\varepsilon$ where $\varepsilon$ is the error bound you have. Then $n=3$ is the degree of the Taylor polynomial that satisfies that error bound, so put $x=0.2$ into $P_n$ since that is what approximates the function, while $R_n$ tells you how good is the approximation of $P_n$.

You'll get $$\sin(0.2)=0.198669\ldots\simeq P_3=0.1986666\ldots$$