m

✓lsin.

m

2(m−1)*+x

Of the Development of Sines and Cosines, of Multiple Arcs in Powers of the Sines and Cosines of the Simple Arcs.

The developments respecting multiple arcs may be divided into two distinct classes. The first includes all series in which the sine or cosine of a

-1)*+*) multiple arc is expressed in powers of those of the

m

simple arc, and the second those in which a power of the sine or cosine of a simple arc is expressed in a series of sines or cosines of its multiples; to the former we shall devote the present section, reserving the latter for the following one.

The series in powers of the sine, cosine, &c. may be either ascending or descending, and accordingly the several problems into which our analysis resolves itself may be enumerated as follow:

To develope, 1o. cos.mx in ascending powers of sin.mx S cos.x.

2o. sin.mx in ascending powers of cos.mx S sin.x.

3o. sin.mx in ascending powers of

4o. cos.mx in descending powers of sin.mx S COS.X.

5o. sin.mx in descending powers of cos.mx S sin.x.

+1) To develope cos.mx in a series of ascending powers of

The law of which is evident. These conditions, which is the development sought.

0, 1, 2, 3,

being substituted for n in cos.m(4n+1), it will
successively assume different values until the num-
ber substituted for n is equal to the denominator of
m; for this value of n the value of cos.m (4n+1)x
will be equal to that obtained by substituting 0 for
n; and all integers greater than the denominator of
m will in like manner give a constant repetition of
values before obtained by substituting for n values
less than the denominator of m. It follows, there-
fore, that cos.m(n+1) is in general susceptible
of as many different values as there are units in the
In like manner
denominator of m, and no more.
cosm(4n-1) is susceptible of the same number
of values, and therefore the coefficient of S is sus-
ceptible of twice as many values as there are units
in the denominator of m, and a like observation
applies to the coefficient of mS'.

Since S and S' involve no functions of x except
cos.x, the change of x into 2n +x makes no
change in their value, and it follows, therefore, that
for a given value of cos.x the second member of
[2] is susceptible of twice as many values as there
It is therefore
are units in the denominator of m.
necessary to show how cos.mx can have several
corresponding values to a given value of cos.x.
The angle x being changed into 2n'+x, n' being
an integer, makes no change in cos.x, but changes
cos.mx into cos.m(2n'+x), which has twice as
many values as there are units in the denominator
of m. Hence the formula [2] will be more gene-
rally and correctly expressed thus,

cos.m(2n'x+x) = cos.m(4n17). S.

+cos. 1(m-1) (4n + 1)π . mS',

73

where both members have the same number of values, and where the values of the indeterminate integers, n', n are supposed to be less than the denominator of m.

It still remains, however, to show the values of each member which correspond respectively to those of the other. Since the value of each member changes by ascribing different values to the integers n' and n, this question only amounts to the determination of the relation between any two cor

Let x = , and therefore S = 1, S' = 0. Hence

responding values of these integers.

cos.m(2n+1)= cos.m(4n+1)*,

or cos.m(4n' + 1) = cos.im(4n+1)x, Since n and n' are not supposed to receive any value greater than the denominator of m (for all the values of the cosine after that would only be repetitions of former values), this last condition can only be satisfied by

n = n'.

Hence the formula becomes*

cos.m(2n+x) = cos.m(4n+1). S
+ cos.(m-1) (4n+1)′′. mS'

[3].

To develope sin.mx in ascending powers of cos.x. By subtracting the value of z-m from that of zm, the result being disengaged from the imaginary symbols becomes

sin.m(2n+x) = sin.}m(4n+1)′′S + sin.}(m — 1) (4n+1). mS'.

[4].

All the observations in the last proposition are equally applicable here. When the denominator of m is an odd integer there are always two values of an angle x whose cosine is given, which are such that sin.mx will be expressed by only one of the two series in [4].

Another form for the development of sin.mx in ascending powers of the cos. may be established by differentiating the series found for cos.mx. By this process we obtain

msin.m(2n+x)=cos.m(4n+1).

ds

dx

* In clearing the formula [1] of imaginary quantities, Lagrange has fallen into an error which was lately detected by Poinsot, and the difficulty explained as above. Lagrange's mistake arose from assuming that

which is evidently erroneous, since the
the second member has but one value.
change on

it does produce a change on

(√— 1)m)=cos.mπ+/-1 sin.¿mæ,

first member has as many different values as there are units in the denominator of m, and He forgot to take into account, that while the change of x into 2n+x produces no

(cos.x+1 sinx)m,

cos.mx+-1 sin.mx.

In fact, without this consideration, Moivre's formula itself is involved in the absurdity of one member having a greater number of different values than the other.

VOL. XVIII. PART I.

M*

the problem will be solved by obtaining the develop a series of ascending powers of the tangent of the ment of zm in ascending powers of y.

Lagrange, and all the mathematicians after him, have fallen into an error in the determination of these coefficients. Poinsot has lately corrected it.