cases, to calculate a few of their terms, to obtain the sine or cosine of the arc required.

69. Several other formulæ for expressing the sine, cosine, &c. in functions of the arc, may also be obtained by means of certain imaginary factors, which have been found of considerable use in physical astronomy, and various other branches of the modern analysis.

sin x
√

Thus, radius being supposed = 1, we shall have cos' x + sin x = 1, of which the first member of the equation is the product of the two imaginary factors cos x + sin x √ 1 and cos x 1. And if any two similar factors cos x + sin x 1 and cos y siny -1 be multiplied together, their product will be cos x cosy sin x sin y + (sin x cos y + sin y cos x) √1 = cos(x+y) + sin (x+y) √ −1.

"

In like manner, (cos x + sin x √ −1) × (cos y + sin y ✔―1) X (cos z + sin z √ −1) = cos (x+y+z) +sin(x+y+2)-1, &c. each of which are like the simple factors, and are produced in a similar way with logarithms, by barely adding the arcs.

And if the arcs x, y, z, &c. be supposed equal to each other, we shall have (cos + sin x-1)= cos 2 x + sin 2 x √-1, and for the three factors (cos x + sin x √ − 1 )3 = cos 3 x + sin 3x √-1. Consequently, in general,

-

(Cos x + sin x −1)" cos n x + sin n x √ — 1. Hence, by transposition and division, we shall obtain the two following equations for the sine and cosine of any multiple of the arc x, in terms of the arc.

Cos n x =

Sin n x =

(cosx+sin x −1)n + (cos t–sin r V−1)*

2

(cos + sin x-1) (cos x-sin x-1)

2-1

70. The sine, cosine, &c. of any arc, or multiple of that arc, may also be readily derived from the wellknown exponential expression e2=1+++

1.2

+1234, &c. where e is the number whose hy

1.2.3 1.2.3.4'

perbolic logarithm is 1.

For, if in this formula, x-1 and -x-1 be successively substituted for z, we shall have

And if these be added to and subtracted from each other, the results, after being divided by 2, and 2−1,

will give

Of which series the second members are the known values, as before found, of the cosine of x and sine

Or, if each of the terms of the numerator and denominator of the second members of the two last equa

And, if mx be substituted for x, in each of these formulæ, we shall obtain the sine, cosine, &c. of any multiple of those arcs: thus, Cos m x =

e

+e
2

2mx√-1

Tan mx=

;

sin m x =

2m x√-1

e

+

2mx√ -1

e

2mx√ + 1); cot mx = √ −1 (°

71. The same formulæ also give e

= COS X

whence, by division, we shall have e

logarithms of each member, 2x-1=log (+/-)

1-tan x-1

But log (1) = 2 (z + ÷ ≈2 + ÷ z°3 + + z' &c.); therefore, putting tan x-1 in the place of z, and dividing each member of the equation by 4 -1, we shall obtain

Which is the known series for the value of any arc in terms of its tangent.

72. It may here also be shown, that any trigonome

trical expression, of the form tan x =

m sin x

1+m cos z

can be converted into a series of the various multiples of the sine of z, taken progressively.

For, if in this equation, there be put instead of the tangent of x and the sine and cosine of z, their values in exponentials (art. 70), we shall obtain

e

+ e 2+ m (ex,

And, consequently, by reduction,

1-mex√-1

-X

1+me

=

or

1-mex√-1

Whence, by taking the logarithm of each member of the equation, and converting the second into a series, according to the form log (1+ z) = z — -z &c. we shall have 2x √-1=

me

x-1 m2 2x√-1 m3 3x√-1 m2 4x√-1

-me

+

e

+ e

4

= 2 sin 2x √ — 1 &c. (art. 70); whence, dividing each side of the equation by 2-1, we shall

x=msinx - sin 2x-msin 3x-sin 4 x - &c.

The former of which series answers to the case tan

m sin z
1+m cos x

73. Several other expressions for the sine, cosine, &c. of multiple arcs, may be derived from some or other of the preceding formnlæ; the most curious and useful of which are the following:

Sin z = 1 sin z

Sin 3 z = 4 sin z sin (— — z) sin ( + z)

3

3

Sin 4 x = 8 sin z sin (— — 2) sin (+2) sin (27 ·%)

4

4

Sin 5 ≈ = 16 sin z sin (— — ≈) sin (7+x) sin (2

5

&c.

Or, generally,

-z)

Sin n z = 2”—1 sin z sin ( − 2) sin ( + z) sin

n

π

x) sin (+x) &c.

n

n

n

Where the series must be continued to as many fac

tors as there are units in n.

74. Cos z = sin ( - )

z

Cos 2 x = 2 sin (− 2) sin (— + z)

(1) The simple series x = sin x - sin 2x + sin 3 x — sin 4 x, &c. of which that given above is a more general form, was first discovered by Euler; as was, also, the series x = sin x sec x sec 1 x secx, &c.

For the algebraical solutions of several of the cases of spherical triangles, according to this form, see Tables Trigonométriques de Borda; where the following theorem is likewise given for the decimal m sinx