n is here a whole number, but, in order to show that the formula holds

when the exponent is a fraction, put a =

n

m

A; then, by this formula,

(cos. a± sin. a.√=1)m=cos. ma ± sin.m A .

cos. n A ± sin. n A.√=1=(cos. A ± sin. A . √—1) " ;

therefore, extracting the mth root of the first and last members, and restoring the value of a, we have

which is the formula of De Moivre.

If we take the reciprocal of each side of this equation we shall have

and if we multiply both numerator and denominator of the first member

n

COS. A sin. A. √—1— (cos. A ± sin. A . √−1) ̄ ~ (17);

m

n m

n

so that the formula (16) remains true, whether be positive or negative.

n

m

If in (16) we make — negative, the signs, in the first member,

m

will be inverted as here, because the sign of the sine is the same as that of the arc.

It may seem to the student that there is a want of generality in the first members of (16) and (17), which ought to contain m values, seeing that the mth root appears in the second members. But this defect is only apparent; for it must be remembered that while the lines sin. A, cos. A, in the second member have each a certain fixed value, the arcs

A, to which these lines indifferently belong are innumerable. The first member involves a proposed fractional part, not of any particular one of these arcs, but of any one of them indifferently; it is easy to see, therefore, that the first member involves a variety of values, and they may be shown to be in number m. (See note A at the end.) (30.) Let the first side of (15) be developed by the binomial theorem and the equation will become

cos." An cos, n—1 Ap+

n (n- ∙1)

cos."-2 Ap2 &c.

=cos. n A ± sin. n A .√—ī;

p being put for the imaginary sin. A.

Now as in any equation the imaginaries on one side are together equal to those on the other, (Alg. p. 126,) we have by expunging all the imaginaries on both sides, the following expression for cos. nA, viz.

In like manner, by expunging all the rational terms on each side of the same equation, and then dividing by 1, there results for sin. n A,

sin. n An cos. n-1 A sin. A

n (n − 1) (n-2)

cos. n-3 A sin.3 A

2.3

+ &c.

From these two expressions may be obtained series, for the value of

the sine and cosine of an arc in terms of the arc itself. For let n=

1

2

finite quantity r; hence

by means of which we may calculate the values of the sine and cosine of any arc x, in parts of the radius or linear unit, when we know the length of r itself, according to the same scale. The length of any arc in parts of the radius is easily ascertained from the known value of 180°, or of a semicircle, in those parts, which, by putting π for the semicircumference to radius 1, is (see Geom. p. 133)

π3.14159265358979, &c.

Ꮖ

π

90

2

so that the length of an arc of r degrees is 180

As in calculating the sines and cosines x may be always taken less than

which series are now in a form suited to immediate calculation. Suppose, for example, the sine and cosine of 1' are required, then

and from knowing the value of sin. 1' and cos. 1' we might compute the sines and cosines for every minute in the quadrant, by means of the formula (3), which, when B=1', becomes

in which A is to be made successively equal to 1', 2', 3', &c. But we shall not enter into the details of this computation here, our present object being to deduce formulas for the sines, cosines, &c. of multiple arcs.

From the general expressions already given for sin. nA, and cos. nA, those for tan. nA, cot. nA, &c. may be readily obtained by help of the equations at (9); we shall not, therefore, occupy the space by writing them down, but confine ourselves throughout the remainder of this article entirely to the consideration of double arcs, as formulas for these are in much more frequent request than for any higher multiple. The formulas of which we speak may, of course, all be deduced from the general expressions investigated in the beginning of this article, but, for simplicity sake, we shall go nearer to first principles, and deduce them from the expressions in art. (26).

Referring to the equations (1), (2), art. (26), we have when A =B,

cos. 2 A= cos.2 Asin.2 A, or cos. 2 A2 cos.2 A — 1,

or cos. 2 A=1-2 sin.2 A

and from the last two of these we immediately get

from which we get two new expressions for cos. 2A, viz.

If instead of A we write 45°-A, then since cos. (90°-2A) = sin. 2A, we have

It may be worth while to remark that the radical in the above ex

pressions for tan. A, cot. A, may be removed by multiplying the numerator and denominator of each fraction by its numerator:

we thus

have

tan. A =

cos. 2 A sin. 2 A

cot. A =

1 + cos. 2 A
sin. 2 A

For the tangent and cotangent of a double arc we have, by division, (18), (19),

that is, dividing numerator and denominator of the second member by

cos.2A, or by sin.2A, and recollecting that

sin.

1

tan., and that

COS.

tan.

which expressions also immediately come from the values of tan. (A + B), cot. (A + B), at (26), by putting A =B.

Comparing the above value of tan. 2A with the expression (8), art. (26), we have,

2 tan. 2 A tan. (45° + A) — tan. (45° — A);

or which is the same thing,

(V).

2 tan. A tan. (45° + 1⁄2 A) — tan. (45° — § A) Formulas for the secants and cosecants of double arcs are easily de

1

duced from those for the cosine and sine, because sec. =