While, on the other hand, dividing the first values of sin A, cos A (formula N) one by the other, there will result,

25. Pursuing an analogous process to that by which we obtained tan (AB), the following may be deduced from formulæ (c): viz.

1+tan A tan B

(c) for the sine of

26. Taking again the equations the sum and the difference, there results, by add. sin (A + B) + Sin (A by sub. sin (A + B) − sin (A The first of these formulæ

-B

B)

divided

B)

2 sin A COS B 2 sin B cos a. by the second

If, therefore, we suppose, A + B = A', A − B = B', = = } (A′ + B′), B = = } (A' — B′), we shall have

whence

But, it has been seen (formula 2) that

The preceding equation, therefore, establishes the truth of the following, viz.

a + b tan † (A' + B′)

=

a- b tan (A

B')

(3.)

27. Similar computations with the equations (c), combined 2 and 2, would produce a great variety of other formulæ. From these, however, since the ingenious student may investigate them nearly as rapidly as he can write them down, we shall only select the following for insertion here.

sin A + sin B = 2 sin } (A + B) COS ¦ (A sin A sin B2 sin § (a

B)

B) COS

(A

(A + B)
B)

(v.)

COS A+COS B = 2 cos 3 (A + B) COS

COS B

COS A2 sin

(A + B) Sin § (A Making A = 90° in the first two, and в others, there result,

1 + sin B2 sin (45° + B) cos (45° —

(45° + B)

1- sin B = 2 sin2 (45° — B) = 2 cos2 (45° + 1B) 1 + cos a = 2 cos2 } /

28. Some useful theorems in the computation of sines, cosines, &c. result from the introduction of the imaginary quantities that arise from the solution of the equations A2 + 1 = 0, and в2 + 10: so that, though these roots, A = ±√1, and B√ 1, are in

5

dications of an absurd supposition, they still serve as convenient instruments of investigation. Their utility must, therefore, be briefly shewn.

The expression sin 2A + cos 2A R or 1, is resolvable into the imaginary factors,

(cos A+ sin A − 1). (cos a

sin A√ 1) = 1.

In like manner, introducing another arc в, and employing imaginary factors, we shall have (cos A+ sin a √1). (COS B + sin B √√ — 1)

COS A COS B

- sin A

sin B +1 (cos A sin B + sin A COS B) = COS (A + B) sin (A+B) 1, [by formula c]; also (cos Asin A 1). (COS B- sin B1) = = cos (A + B) - sin (A + B) √1. By a like calculation it will be seen that (cos Asin A√ 1). (cos B± sin в √1) (cos csin c 1) cos (A + B + C) ± sin (A + B + C) √1. It appears, therefore, that the multiplication of this class of quantities is effected by simply adding the arcs, a property analogous to the characteristic property of logarithms.

29. If the arcs A and B be supposed equal and two imaginary factors be taken, we shall have (cos Asin A √1)2= cos 2A + sin 2A -1; if three factors be taken, the result will be (cos a ± sin a √ — 1)3 == COS 3A sin 3A ✔ - 1.

Whence, in general,

(COS Asin A√ − 1)" = cos na ± sin na √ − 1. 30. From the last equation we obtain sin na √ -1= (cos Asin A √ — 1)".

cos na,

and sin na -1= · (cos A — sin a √√−1)" + cos na, dividing the sum of these by 2 √ 1, there results the following expression for the sine of any multiple arc;

viz.

sin na

(cos A+ sin A √ — 1)" — (cos A—sin A √ — 1)"

31. These quantities may readily be expanded into series, by means of the binomial theorem; in which case all the terms affected with imaginaries will be annihilated, and there will be obtained the two following general series, one for the sines the other for the cosines of multiple arcs.

n (n - 1) (n-2) 22 3

1.2.3

COS

A

sin 3A+

1.2.3.4.5

(x.)

1.2.3.4.5.6

n

- 6

A sin A+ &c.

32. If the arc A be supposed indefinitely small, then will sin AA, cos A = 1, and that the arc na may become an assignable quantity, n itself must be regarded as indefinitely great; in that case unity will (practically) - 1, n 2, &c. so that n (n − 1) vanish in the terms n1) (n = n3, &c. Hence, will become n2, n (n -2) if we put na = c, we shall have (since sin a = A, and

C2

A =

=), sin 2A =

n

sin 3A = &c. while cos A, cos2

A, cos 3A, will each 1. The two last formula will, therefore, be reduced to the following, by means of which it will be easy to express the sine and cosine of any arc, in parts of that arc, or in decimals of the radius, that is, to calculate the natural sine and cosine of such

These series will converge rapidly when the are is small.*

33. From expressions, such as the preceding, for the sines, tangents, &c. of the sums, differences, and products of two arcs, or angles, it would be easy to pass to those for three, four, or more arcs. But as the properties which might thus be developed, however curious and elegant, are comparatively of little utility; we shall not present them here, but confine our investigation, either to the angles of a plane triangle, or to those which have an obvious relation to them.

A, B, and C, being the three angles of a plane triangle, since c = 180° — (A + B),

tan Atan B+ tan C tan A tan B tan ċ.... · (4.) Dividing this equation by the whole of the first mem ber, and substituting for the products of the tangents divided by their sum, their corresponding values in cotangents (from equa. s), there will result,

(5.)

1 cot A cot B + cot в cot c+cot A cot c.... If the sum of A, B, and C, instead of being 180°, were 360°, the same formulæ would result, as is evident from the consideration, that in that case also we should have tan c = tan (A+B). It may, farther, be readily seen that the same expression is applicable, so long as A+B+Cn. 180° or 2n.90°.

*This branch of the subject is pursued, with great elegance, to a considerable extent, by Euler, in his valuable work, Analysis Infinitorum.

360°,

+ The property restricted to the case when A + B + C = was announced and demonstrated by Dr. Maskelyne in the Phi losophical Transactions for 1808: in the case of the triangle it had been previously demonstrated by Cagnoli,