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dications of an absurd supposition, they still serve as convenient instruments of investigation. Their utility. must, therefore, be briefly shewn.” Theore sion sin *A + cos ?A = R or 1, is resolvable into the imaginary factors, (cos A + sin A V-1), (cos A – sin A M – 1) = 1. In like manner, introducing another-arc B, and employing imaginary factors, we shall have (cos A + sin A A/- 1). (cos B+ sin B v — 1) = cos-A cos B – sin A sin B + v — 1 (cos A sin B + sin Ascos B) = cos (A + B) + sin (A + B) v — 1, [by formula cl; also (cos A. — sin A v – 1). (cos B-sin B v — 1) = cos (A + B) — sin (A + B) v — 1. By a like caleulation it will be seen that (cos A it sin A V – 1). (cos B it sin B V-1): . (cos c + sin e v — 1) = cos (A + B -- c), it sino (A + B + C) v 1. It appears, therefore, that the multiplication of this class of quantities, is effected bysimply adding the arcs, a property analogous, to the characteristic property of logarithms. 29. If the arcs. As and B be supposed, equal and two imaginary factors be taken, we shall have (cos A + sin A A/,-1)* = cos 2A, FE sin 2A.V. — 1 ; if three factors be taken, the result,will be (cos A, it sin A V-1)3 = cos 3A-Esin 3A v ––1, Whence, in general, (cos A + sin A v — 1)" = cosm A =E sin na V - 1. 30. From the last equation we obtain sin na w/ - 1 = (cos A.-- sin A V – 1)" — cos n A, and sinn A V-1'='-(cos A – sin A V– 1)" -- coson A, dividing the sum of these by 2 v — 1, there results the following expression for the sine of any multiple arc;

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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. I

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A, cos 3A, will each = 1. The two last formulae 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 arC,

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These series will converge rapidly when the arc 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 A + tan B tan c = — tan (A + B) = T T-tan Atan no whence, taking away the denominator, s" s tan c – tan A tan B tan c = - tan A — tan B, and, by transposition, tan A.-- tan B + tan c = tan A tan B tan c..... (4.) Dividing this equation by the whole of the first member, and substituting for the products of the tangents divided by their sum, their corresponding values in cotangents (from equa. s), there will result, 1 = cot A cot B + cot B cot c + cot A cotc. ... (5.) If the sum of A, B, and c, instead of being 180°, were 360°, the same formulae would result, as is evident from the consideration, that in that case also we should have tan c = — tan (A + B).t . It may, farther, be readily. seen that the same expression is applicable, so long as A + B + c = n. 180° or 2n.90°.

* This branch of the subject is pursued, with great elegance,

to a considerable extent, by Euler, in his valuable work, Ana

lysis Infinitorum. + The property restricted to the case when A+ b + c = 360°,

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. . . . ; - D 2

*

, 34. Suppose. A + b + c = 90°, then tan c = cot

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cluded angle c-given, the logarithms of those sides are given; as frequently happens in geodesic operations, rand in astronomical tables for-the-distances of the planets from the sun. Here if a and b are regarded as the sides of a right angled triangle, in which a denotes the angle opposite to the side a, we shall have

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