Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/263

Hence for all values of $x$ in the neighborhood of some fixed value $a$ we have the approximate formula

(G)	$\sin x=\sin a+\cos a\left(x-a\right)$

Transposing sin a and dividing by $x-a$, we get

${\frac {\sin x-\sin a}{x-a}}=\cos a.$

Since $\cos a$ is constant, this means that:

The change in the value of the sine is proportional to the change in the angle for values of the angle near $a$.

For example, let $a=30^{\circ }=.5236$ radians, and suppose it is required to calculate the sines of $31^{\circ }$ and $32^{\circ }$ by the approximate formula (G). Then

$\sin 31^{\circ }$	$=\sin 30^{\circ }+\cos 30^{\circ }(.01745)$[1]
	$=.5000+.8660\times .01745$
	$=.5000+.0151$
	$=.5151.$

Similarly, $\sin 32^{\circ }=\sin 30^{\circ }+\cos 30^{\circ }(.03490)=.5302$.

This discussion illustrates the principal known as interpolation by first differences. In general, then, by Taylor's Series, we have the approximate formula

(H)	$f(x)=f(a)+f'(a)(x-a).$

If the constant $f'\left(a\right)\neq 0$, this formula asserts that the ratio of the increments of function and variable for all values of the latter differing little from the fixed value a is constant.

Care must however be observed in applying (H). For while the absolute error made in using it in a given case may be small, the percentage of error may be so large that the results are worthless.

Then interpolation by second differences is necessary. Here we use one more term in Taylor's Series, giving the approximate formula

(I)	$f\left(x\right)=f\left(a\right)+f'\left(a\right)\left(x-a\right)+{\frac {f''\left(a\right)}{2!}}\left(x-a\right)^{2}.$

1. $x-a=1^{\circ }=.01745$radian.