by the theorem for the tangents of the sums and differences of arcs. For, when the radius is unity, the tangent of 45° is also unity, and tan (A + B) will be denoted thus: 1 + tan B And this, again, may be still further simplified in practice. 23. The secants may readily be found from the tangents by addition. For (prop. 3) sec A = tan A + tan A comp A. Or, for the odd minutes of the quadrant the - I secants may be found from the expression sec = cos' os Other methods for all the trigonometrical lines are deduced from the expressions for the sines, tangents, &c. of multiple arcs; but this is not the place to explain them, even if it were requisite to introduce them at large into a cursory outline. CHAPTER III. Solutions of the several Cases of Plane Triangles. r 1.THERE being in every plane triangle, six things, namely, three sides and three angles, of which some three are given to determine the other three; and the combinations that can be formed out of six quantities, - 6.5. taken three and three being = 1 - 2 : first sight be imagined that 20 distinct rules would be required in this branch of trigonometry. But though the varieties of data are in truth thus numerous, it will or 20; it might at * soon appear, that the number of cases which require distinct rules, are very few. Let A, B, C, denote the angles of a plane triangle, and a, b, c; the sides respectively opposite; then the twenty varieties of data will be these, viz. (3) abc... (1) abA ... (1) abe ... (2) abo 1) acA ... (2) acB . . . }} acco 2) bcA ... (1) bcB ... (1) bcc (4) ABc... (1) Aba ... (1) ABö ... (*) ABc }:::::: *) Acó ... (1) Acc (*) Bca ... {i} BCô ... (1) Bcc. Here the varieties which are marked by the figure 1, have this in common, that a side and its opposite angle are two of the given parts; and, if it be considered that when two angles of a plane triangle are known, the third is, in fact, given, because it is the supplement of their sum, it will appear that those varieties which are marked with an asterisk fall under this first case: so that the varieties comprehended in this case are fifteen. Of the remaining five, three, indicated by the number 2, have this in common, that two sides and an included angle are given; they, therefore, constitute a second case. The example marked 3, has three sides given, but no angle; this makes a third case: and the remaining variety in which the three angles are given, but no side, would make a fourth case, were it not (see chap. i. 6) that for want of a side among the data, the problem thus expressed is unlimited. ... Our twenty varieties, therefore, only furnish three distinct cases, to the solution of which we shall now proceed. CASE I. 2. When a side and its opposite angle are among the given parts. Here the solution may be obtained by means of chap. ii. prop. 14, where it is demonstrated that the sides of plane triangles are respectively proportional to the sines of their opposite angles. C In practice if a side be required, begin the proportion with a sine, and say, "As the sine of the given angle, To its opposite side; . So is the sine of either of the other angles, To its opposite side. If an angle be required, begin the proportion with a side, and say, . As one of the given sides, Is to the sine of its opposite angle; So is the other given side, To the sine of its opposite angle. The third angle becomes known, by taking the sum of the two former from 180°. Note 1. Since sines are lines, there can be no impropriety in comparing them with the sides of triangles; and the rule is better remembered by young mathematicians, than when the sines and sides are compared each to each. Note 2. It is usually, though not always, best to work the proportions in trigonometry by means of the logarithms, taking the logarithm of the first term from the sum of the logarithms of the second and third, to obtain the logarithm of the fourth term. Or, adding the arithmetical complement of the logarithm of the first term to the logarithms of the other two, to obtain that of the fourth. Note 3. It is an excellent plan to accustom the pupil to draw (previously to his commencing the computations in plane trigonometry) not a rough, but a meat and accurate, sketch of the triangle proposed, from the given data, by means of scale, protractor, and compasses. Such construction will enable him at once to trace the peculiarities of the problem, and to detect its ambiguities, if there be any. It will also, if accurately performed, upon a scale of moderate size, give the sides to within their 200th part, and the angles true to within half a degree. o Note 4. The triangle being carefully constructed, and marked by suitable letters of reference, as A and B for example, at the extremities of the base, c at the vertex, then, according to the nature of the problem, write down the requisite proportion in four distinct lines, with the letters of reference to each term, and the given numbers to the three first: o these numbers place their respective logarithms; find the logarithm to the fourth term by the directions in note 2, and ascertain its angular or lineal value, by means of the tables. *** The last three notes are not restricted to the present case, but extend in their application to the usual practice of plane trigonometry. o : - Example I. 3. In a plane triangle are given two angles equal to 58°7', and 22° 37', respectively, and the side between them 408 yards. Required the remaining angle and sides, Construction... On an indefinite right c line, set off, from a convenient diagonal scale, the distance AB = 408. From the point A draw a right line Ac, to make A with AB an angle of 58° 7'; and from the point B another line, turned towards the former, and to make with BA an angle of 22° 37'. The intersection cy of these two lines determines the triangle; and the sides Ac, Bc, measured upon the scale of equal parts from which AB was laid down, are found to be 159 and 351 respectively. Computation. Two of the angles being known, their sum 80° 44' taken from 180°, the sum of the angles in a plane triangle, leaves 99° 16' for the third angle c. This, being an obtuse angle, its sine is to be found in the table by taking that of its supplement 80°44', which (chap. i. 19) is the same. Hence, Ac and Bc, therefore, are 351-02 yards, and 158.98 yards, respectively. 4. In the preceding operation, instead of adding together the logs. of the second and third terms, and subtracting that of the first from their sum, the work has been performed thus:–The right hand figure of the upper line was taken from 10, and each of the other figures from 9, and their several remainders added to the numbers below them in the respective columns. This is easily effected in practice by making those remainders emphatical in adding downwards. Thus, in the operation for Bc, we begin at the right hand, and adding downwards say, ten and 2 are 12, and 8 are 20; set down 0: carry 2, and four are 6, and I make 7: nothing added to 6 and 7, gives 13; set down 3: carry 1, and seven are 8, and 6 are 14, and 9 are 23; set down 3: carry 2, and five are 7, and 8 are 15: and so on, to the left hand column, which added in the same way amounts to 12; of which the 2 are put down, and the 10 rejected, to compensate for what has been borrowed in the process by the arithmetical complement. This method is very easy in practice, and is found less liable to produce error than that in which the arith. comp. is put down at once from the tables. Example II. 5. In a plane triangle ABC are given Ac = 216; cB = 117; the angle A = 22° 37'; to find the rest. Construction. Draw an indefinite right line ABB', |