204. The error of employing the differentials in any case may be determined approximately by developing the equation of finite differences and comparing it with the corresponding differential equation. We shall select a simple example. We have from (387) and its corresponding differential equation in (399)

or substituting

sin AB =

AB sin 1", sin AB = AB sin 1", and also B for BAB in the second term, which will affect so small a term but slightly,

Comparing this with the differential equation above, the error of employing the latter is approximately

It appears from this example that the error is expressed by a term involving the square of the increment; and if we develop all the equations of finite differences we shall find that they differ from the corresponding differential equations by terms involving the squares and higher powers of the increment. Hence, employing the differentials instead of the finite differences amounts to neglecting the terms involving the squares and higher powers of the increments.

205. The differential relations above obtained could have been deduced more directly from the formulæ of plane triangles by differentiation, employing the values of the differentials given in Art. 192. Thus in CASE I, A and c being constant, if we differentiate the equation

we have

ac sin A cosec C

dac sin A d cosec C

= - c sin A cot C cosec CdC

a cot Cd C

as in (398).

The student may exercise himself by deducing the other relations of (398), (399), and (400) in a similar manner.

CHAPTER XIII.

TRIGONOMETRIC SERIES. DEVELOPMENTS OF THE FUNCTIONS OF AN ARC IN TERMS OF THE ARC, AND RECIPROCALLY.*

206. THE investigation of trigonometric series is most readily carried on with the aid of a few elementary principles of the Differential Calculus. All that will be required here will be no more than is generally given in the first chapter of a treatise on that subject, namely, the differentiation of simple algebraic functions, and Taylor's Theorem. We shall employ the following expression of this theorem: d.fy h d2.fy h2 d3.fy h3 + dy 1

f(y+ h)=fy +

+

+ &c. (402) dy2 1.2 dy3 1.2.3

in which fy denotes what f(y+h) becomes when h = 0 and

d.fy d2.fy

&c., are the successive differential coefficients, or dedy' dy' rivatives of fy.

207. To develop sinx and cosx in terms of x.

We shall first develop sin (y + x) and cos (y + x) by (402). By (361) and (362), if

*The leading results of this Chapter being of very general utility and constant application are printed in the larger type, but as they are not referred to in the subsequent large print of this work, and moreover require a limited acquaintance with the Differential Calculus, the student can omit them at the first perusal, and pass directly to Part II.

so that the values of the coefficients of the series (402) recur in the - cosy, and therefore f(y + x)=

order sin y,+ cos y, sin y, +

the coefficients will recur in the order + cos y, + siny, and (402) will give

If now we put y = 0 in (403) and (404), sin y = 0, cos y = 1, the alternate terms of the series vanish, and we have

It may be observed that (406) can be deduced from (405) by dif ferentiation.

208. The series (405) and (406) are directly available for the construction of the trigonometric table. For this purpose x in the series must be expressed in arc, since (361) and (362), upon which the preceding demonstration rests, require x to be in arc, Art. 9.

EXAMPLE.

Find cos. 10°. Reducing 10° to arc, by Art. 9, we have

and computing separately the positive and negative terms of (406),

agreeing with the tables, which give 9848078. The student may, for practice, verify any other sine or cosine of his table.

209. To develop tan x in terms of x.

Representing the coefficients in the series (405) and (406) by letters, we have

If we perform the division of the numerator by the denominator, we perceive that the result will be a series containing only odd powers of x, and commencing with the term x. But as the law for the successive formation of the coefficients is not easily shown in this way, we shall resort to the following process. Assume the series to be tan x c1x + c3 x3 + c ̧ Ñ3 + &c.

and differentiate it; we find, by (363), after dividing by dx,
1+tan x = c1 + 3 c2 x2 + 5c, x2 + &c.
or, since from the division of (407) we know that c1 = 1,

tan2 x = 3 c, x2 + 5 c2 x2 + 7 c1 x2 + 9 c, x2 + &c.

(m)

(n)

where the law of derivation is obvious. We have preserved the factor c1, although it is equal to unity, in order to render this law more apparent.

Since the first and last terms of these expressions are equal, as also the terms equally distant from them, we may write them as follows:

the succeeding terms evidently involve only the odd powers of x.

Therefore let

(0)

The coefficients cannot be determined by the method of the preceding article in consequence of the negative exponent in the first term; but they are directly deducible from those of the series for tan x. We have by (142)

tan x cot x - 2 cot 2 x

(d)

Now the series (0) being true for any value of x will give cot 2 x by substituting 2 x for x, whence

Subtracting this from (0) we have by (p)

tan x = (2a — 1) d ̧ x + (2a — 1) d, x3 + (2o — 1) d, x3 + &c. Designating the coefficients of (408) by c1, C2, C., &c. we have also

* Euler, and after him Cagnoli and others, make these coefficients depend upon those of the series sin x and cos x, but the number of given quantities by which each coefficient is expressed is double the number required in the method of the text.