(73.) To find the numerical value of the sine of 1'. 2 (cos. A)=1+ cos. 2 A by Art. (43.) Now, we should find exactly in a similar manner- The first of these three angles is to the second, and the second to the third, in the ratio of 2: 1; and it will be observed, by inspection of the above values, that their sines are very nearly in the same ratio, and, indeed, if we took only 8 places of decimals would appear exactly so. Hence we conclude that, when the angles are very small, we may consider, without sensible error, that the sines are proportional to the angles. Hence to find the sine of 1' we have sin. 1.0002556254 :: 60: 52" 44"" 16, whence sin. 1'=.0002908882 * * Comparing this value of the sine of 1' with the numerical value of an angle of one minute, found in the example to Art. (10.), where the angular unit is that mentioned in Art, (7.), it appears that those values coincide for at least ten places of decimals. ..sin. (A+B)=2 sin. A sin. (A-B) 4 sin. A :( Let B l'; and put A successively equal to 1', 2', 3′, &c., then sin. O') sin. 1') sin. 2') - 4 sin. 1' (sin. 30") The sine of 30" is known immediately from the sine of l′; and therefore the value of 4 (sin. 30") is easily computed. This being done beforehand, and having tabulated the values of the sines of 1', 2′, 3′, &c., n', then to find the sine of (n+1′), take the last tabulated value, add to it the difference between that and the preceding one, and from the result subtract the last tabulated value multiplied by a constant quantity, the value of 4 (sin. 30"). If we wished to calculate successively the values of the series 2o, 3o, &c., knowing those of 1° and of 30', the same formula will evidently apply. (75.) The values of the cosines may be calculated from those of the sines, or they may be found as follows: since (76.) In this manner the sines and cosines of all angles may be computed. It is not necessary, however, to carry the operation farther than for 45°, since the sines of all angles less than 45° would give the cosines of their complements, and the cosines of all such angles the sines of their complements. (77.) After we have proceeded as far as 30°, the labour of calculation may be considerably abridged by the following formula: Sin. (A+B) 2 sin. A cos. B- sin. (A-B), which gives, making A 30° (since sin. 30° = ). Sin. (30° + B) = cos. B 2', 3', &c. successively. sin. (30°- B), and putting B equal to l' Whence the values of the sines of all angles may be formed from 30° to 45°, by simply taking the difference of previously tabulated values. which formula possesses the same advantage as the previous one. (79.) The values of the sines and cosines of all angles at any given interval, as far as 90°, having been found, the values of the tangents of the same angles may be obtained by dividing that of the sine by that of the cosine. Having thus calculated as far as 45°, the labour of computation may be abridged by the following formula: tan. (45°+B) = 2 tan. 2 B + tan. (45°-B) tan. 45° 1' 2 tan. 2' + tan. 44° 59' Whence the values of all angles between 45° and 90° are determined by means of values previously tabulated. (80.) The values of the cotangents of angles are the same as those of the tangents of their complements, and are therefore known directly from what precedes. Also, 1 1 sec. A= ; cosec. A cos. A sin. A whence the values of the secants and cosecants may be computed. (81.) When the values of the functions of successive angles are calculated from the previously tabulated values, it is manifest that any error in one value will extend to the succeeding ones. To obviate this danger it is necessary to make use of formulæ of verification, by which the value of any function is calculated by some independent method. The agreement of the value thus found, with that obtained by the first-mentioned method, is the test of accuracy. Any of the methods used in the first part of this section might be applied for this purpose. The formulæ, sin. A = sin. 2 A} {√1 + sin. 2 A√1 Art (44.) cos. A {√1 + sin. 2 A±√1 sin. 2 A} are convenient, as well as that which is frequently termed Euler's formula of verification. It is this: - sin. A + sin. (72°+A)—sin. (72°~A)=sin. (36°+A)—sin. (36° — A). To prove this formula, we have sin. (72°+A)—sin. (72°—A) — 2 cos. 72° sin. A Art. (38.). Also sin. (36°+A)-sin. (36°-A) = 2 cos. 36° sin. A = 2(1–2 (sin. 18°)2 } sin. A * This formula is easily found (5-1)) =2(1–2. sin. A 16 tan. (45° + B) = 1-tan. B 1-tan. B tan. (45° - B) 1+tan. B From these values the truth of the formula is evident. Another formula of the same kind, called Le Gendre's formula, may found in a similar manner. It is be cos. A sin. (45°+A)+ sin. (54°-A)-sin. (18°+A)-sin. (18°—A). (82.) The above methods enable us, without any very serious labour, to calculate the values of the trigonometrical functions of angles to any required degree of accuracy, and to a greater number of decimals, if required, than are usually found in trigonometrical tables. The labour of calculating these values for a series of consecutive angles is also very much diminished by the method of interpolation, which requires a knowledge of a higher analysis than the reader of an elementary treatise like the present can be supposed to be acquainted with. The values of any function, as the sine, of two angles at a considerable interval, being calculated by some of the previous methods, the values of the sines of the intermediate angles are computed by this method of interpolation. It has been furnished by comparatively modern analysis, at least it was quite unknown to the first laborious calculators of mathematical tables, as well as other convenient formulæ which the progress of trigonometry has since afforded. Industry, however, supplied the place of more compendious methods, and before the beginning of the sixteenth century, tables of natural sines, tangents, &c. were already computed, very nearly equal in accuracy to those we possess at present. That the student may the more readily understand the arrangement of tables of this description, we shall give specimens of two pages extracted from Hutton's tables. The first of these is the first page of the tables, the second the last page. Sine. Dif. Covers. Cosec. Tang. Cotang. Secant. Vers. D. Cosine. 0 0000000 1 0002909 2 0005818 3 0008727 4 0011636 5 0014544 6 0017453 7 0020362 8 0023271 9 0026180 10 0029089 11 0031998 12 0034907 13 0037815 14 0040724 15 0043633 16 0046542 17 0049451 18 0052360 19 0055268 20 0058177 21 0061086 22 0063995 23 0066904 24 0069813 25 0072721 26 0075630 27 0078539 28 0081448 29 0084357 30 0087265 31 0090174 32 0093083 33 0095992 34 0098900 35 0101809 36 0104718 2909 2909 2909 2908 2909 2909 2909 2909 2908 9938914 163 70325 2909 2909 2909 2908 2909 2909 9909826 110.89656 0090178 110-89205 1.0000286 0000286 1.0000308 0000308 1.0000332 0000332 1.0000356 0000356 1.0000381 0000381 1.0000407 0000407 1.0000433 0000433 1 0000461 0000461 1.0000489 0000489 1.0000518 0000518 1.0000548 0000548 2909 0055269 180-93220 1.0000153 0000153 16 9999847 41 9999831 4. 18 9999813 30 18 9999795 38 24 24 25 26 26 30 9857470 70 160474 0142545 70-153346 1.0001016 0001016 1.0001234 0001234 2909 0151273 66 105473 55 0159982 2908 9840018 62 507153 0160002 62 499154 1.0001280 0001280 9999776 37 9999756 3€ 9999736 35 9999714 34 9999692 23 9999668 32 9999644 31 9999619 30 9999593 29 9999567 2 9999539 27 999951126 9999482 25 9999452 24 32 34 34 25 9999421 9999389 22 9999357 21 9999323 20 9999289 19 9999254 18 9999218 17 9399181 16 9999143 15 9999105 14 9999065 13 9999025 12 163432 9998984 11 9998942 10 9998900 9998856 9998812 9998766 9986 |