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 ( &sect;139). For example, assume

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

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), &sect;145)

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

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

is correct to two decimal places (i.e. error < .01), knowing that $$|f^{(v)}(x_1)| \le 1$$ we have, from (D) and (E),

$$\frac{\left| x^5 \right|}{120} < .01;$$ i.e. $$\left| x \right| <\sqrt[5]{1.2};$$ or $$ \left | x \right| \le 1.$$ 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 $$sin x$$ by Taylor's Series, (62), &sect;144, in powers of $$x - a$$, we get

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