A TREATISE OF PLANE TRIGONOMETRY - Tapa blanda

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1855 Excerpt: ...will be negative. The latter will not terminate in the same quadrant as the other--and the signs of the tabular lines must be accommodated to this circumstance. Thus, the sine of AS will be positive, while that of AS'" will be negative. (Art. 194.) When a greater arc is subtracted from a less, if the latter be positive, the remainder must be negative. (Alg. 40.) TRIGONOMETRICAL FORMULA 206. From the view which has been here taken of the changes in the trigonometrical lines, it will be easy to see, in what parts of the circle each of them increases or decreases. But this does not determine their exact values, except at the extremities of the several quadrants. In the analytical investigations which are carried on by means of these lines, it is necessary to calculate the changes produced in them, by a given increase or diminution of the arcs to which they belong. In this there would be no difficulty, if the sines, tangents, &c, were proportioned to their arcs. But this is far from being the case. If an arc is doubled, its Bine is not exactly doubled. Neither is its tangent or secant. We have to inquire, then, in what manner the sine, tangent, &c, of one arc may be obtained, from those of "otter arcs already known into the cosine of the second + the product of the sine of the second into the cosine of the first. The problem on which almost the whole of this branch of analysis depends, consists in deriving, from the sines and cosines; of two given arcs, expressions for the sine and' cosine of their sum and difference. For, by addition and subtraction, ai few arcs may be so combined and varied, as to produce others of almost every dimension. And the expressions for the tangents and secants may be deduced from those of the sines and cosines. Exp...