Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/262

This is an alternating series for both positive and negative values of $x$. Hence the error made if we assume $\sin x$ to be approximately equal to the sum of the first $n$ terms is numerically less than the $(n+1)$th term ( §139). For example, assume

(B)	$\sin x=x,$

and let us find for what values of x this is correct to three places of decimals. To do this, set

(C)	$\left|{\frac {x^{3}}{3!}}\right|<.001.$

This gives $x$ numerically less than ${\sqrt[{3}]{.006}}(=.1817)$; i.e. (B) is correct to three decimal places when $x$ lies between $+10.4^{\circ }$ and $-10.4^{\circ }$.

The error made in neglecting all terms in (B) after the one in $x^{n-1}$ is given by the remainder (see (64), §145)

(D)	$R={\frac {x^{n}}{n!}}f^{(n)}\left(x_{1}\right);$

hence we can find for what values of x a polynomial represents the functions to any desired degree of accuracy by writing the inequality

(E)	$|R|<$ limit of error,

and solving for $x$, provided we know the maximum value of $f^{(n)}(x_{1})$. Thus if we wish to find for what values of $x$ the formula

(F)	$\sin x=x-{\frac {x^{3}}{6}}$

is correct to two decimal places (i.e. error < .01), knowing that $|f^{(v)}(x_{1})|\leq 1$ we have, from (D) and (E),

${\frac {\left|x^{5}\right|}{120}}<.01;$ i.e. $\left|x\right|<{\sqrt[{5}]{1.2}};$ or $\left|x\right|\leq 1.$

Therefore $x$ gives the correct value of $\sin x$ to two decimal places if $|x|=1$; i.e. if $x$ lies between $+57^{\circ }$ and $-57^{\circ }$. This agrees with the discussion of (A) as an alternating series.

Since in a great many practical problems accuracy to two or three decimal places only is required, the usefulness of such approximate formulas as (B) and (F) is apparent.

Again, if we expand $sinx$ by Taylor's Series, (62), §144, in powers of $x-a$, we get

$\sin x=\sin a+\cos a\left(x-a\right)-{\frac {\sin a}{2!}}\left(x-a\right)^{2}+\cdots .$