These may obviously be extended to other arcs, by means of formulæ (P), (Q), (R), of this chapter, and prop. 3 of chap. ii. 42. Operations of this kind ought not to be carried far without being subjected to checks and proofs. For this purpose, after the sines and cosines are found, the tangents, secants, &c. are easily verified by their mutual relations. The sines and cosines themselves, are examined by means of some of the following "formulæ of verification." sin (18° +A) + sin (18° - A) A A) 2 sin 18° cos A sin A + sin (36° − a) + sin (72° + a) = sin (36° + a} + sin (720s). Here A may have any assigned value less than the least specified arc. Thus, if A were 9°, we should have sin 9° + sin 27° + sin 81° = sin 45° + sin 63°, and if from any table the sines thus taken make equal sums, it is highly probable they are all correct. Taking these from Dr. Hutton's tables, we have sin 9° = •1564345 43. Of the formulæ investigated in this chapter, those which have letters of reference (A), (B), (C), &c. relate to the sums, differences, multiples, &c. of sincs, tan gents, &c. while those which have figures of reference, (1), (2), (3), &c. will be employed in the solutions of plane triangles. Other kinds will find their application in subsequent parts of this introduction; and the student will do well, after he has gone through their investigation, to arrange them in separate tables for use. We shall terminate the present chapter by subjoining three examples. Example I. Given the three sides of a plane triangle 40, 34, and 25, respectively, to find the largest angle, by formula (15). Here (a + b + c) -c (a + b 15.5..log 1·1903317 c) - b = 24.5.. log Their prod. shown by the sum of the logs Also, (a+bc) - a (a + b−c) 1.3891661 2.5794978 9.5.. log = 0.9777236 49.5.. log = 1.6946052 Their prod. shown by the sum of the logs.. 2.6723288 The latter sum taken from the former borrowing 20 for radius squared, gives 19.9071690. The latter log. by 2 for the , gives 9.9535845. This, by the tables, is nearly tan 41° 56′, and by the formula it is tan A; therefore A41°56', and the required angle = 83° 53′ nearly. Note. In chap. iii. case 3, the same result is obtained by a different process. Example II. What are the angles of that plane triangle whose natural tangents are integers? It is evident from equa. (4) that the sum of the three tangents must be equal to their continual product. Now, the only three integers which possess this property, are 1, 2, and 3; which are, therefore, the tangents required. The angles of which they are the tangents, are 45° 63° 26′ 6′′, and 71° 33′ 54′′; whose sum makes 180°, as it ought to do. Example III. There is a plane triangle whose sides are three consecutive terms in the natural series of integer numbers, and whose largest angle is just double the smallest. Required its sides and angles. That the student may compare the two methods, we shall present both a geometrical and an algebraical solution to this problem. 1st. Geométrically. Analysis. Suppose the triangle ABC to be that whose sides CB, BA, À¤, are respectively as three consecutive terms in the increasing series of integer numbers, and the greater angle ABC equal to twice the less angle BAC. From c upon AB let fall the perpendicular CD, make Db = DB, and join BC. Then, because the angle cbв (= ABC) is equal to 2CAB, the points à and c are in the circumference of a circle whose radius is ba, or bc, and centre b. But AB:AC + CB (= 2AB) :: ÁC = Ab (= 4AB AB bDB CB (= 2): AD - DB 4) CB; hence CB is given; and be cause AB CB + 1 = AC 1 (by hypothesis), the three sides are given, to construct the triangle. Construction. Let the right line AB be made equal to 5, on any scale of equal parts; from centres A and B with radii 6 and 4 respectively, describe arcs to intersect each other in c; draw AC, BC, and ABC is the required triangle. Demonstration. The sides CB, BA, AC, are three con-secutive terms in the series of natural numbers (by const.) From c upon AB demit the perp. CD,' set off Db = DB, and draw bc. Then, AB (= 5): ac + Cb (= 6 + 4) :: AC CB (64): AD DB Ab 4. Therefore Ab BC= bc: and hence the points A and c are in the circumference of the circle whose centre is b and radius ba, or bc; and consequently (Euc. iii. 20) angle cóв = CBA = 2 angle CAB. 2dly. Algebraically. Denote BC by x 1, BA by x, and AC by x + 1; then, (BE being perpendicular to AC the longest side) we have AC: AB + BC :: AB BC AE EC, that is, Half this added to half 1 x + 1 x2 + 4x + = 2 2x + 2 2 x2 + 4x x + 4 COS A. rad :: AE: cos a, or, x:1:: Whence sin A = √(1 - COS 2A) and (by the quest.) this angle = 2A. 2x + 2 12) By equating these two values of sin B, we have and x 5. The sides, therefore, of the triangle are 4, The cosines of the angles are, cos A = And the angles are, A = 41° 24′ 34′′ 34′′ c = 55° 46′ 16′′ 18′′ B 82° 49′ 9′′ 8" CHAPTER V. Application of Plane Trigonometry to the determination of Heights and Distances. 1. ONE of the most familiar, and at the same time, useful, applications of plane trigonometry, is to the determination of the altitudes and distances of remote objects, the former usually designated by the term altimetry, the latter by that of longimetry. It is not intended to treat them separately here, nor to treat them jointly much in detail; but simply to present and solve a selection of such problems as are most likely to occur in practice, and as are best calculated to suggest the modes of procedure in other cases. 2. The instruments employed to measure angles are quadrants, sextants, and other circular instruments. For military men, perhaps, the best instrument is a pocket sextant, which, if accurately constructed by a skilful artist, will enable a careful observer to take an gles to within a minute of the truth. But for general purposes the most proper instrument for the measuring of horizontal and vertical angles, is a theodolite furnished with a compass and level, one or two telescopes, and, a vertical arc. Such an instrument, when each circle is 6 or 7 inches diameter, and has a nonius adapted to it, will enable the observer to ascertain angles to half a minute. |