DIFFERENTIAL VARIATIONS OF SPHERICAL RIGHT TRIANGLES. The preceding may also be used for right triangles; but it may be desirable to have the same variables in both members, as in the following formulæ derived from those of Arts. 140, 141, and 142: 152. The differential variations are often employed for approximate results, instead of the equations of finite differences, when the increments are very small. The remarks of Pl. Trig. Art. 203, apply here also, but it is not necessary to introduce the radius in seconds, since all the parts of a spherical triangle are expressed in the same unit. DIFFERENTIAL VARIATIONS OF SPHERICAL TRIANGLES WHEN ALL THE PARTS ARE 153. Let the equation VARIABLE. cos a cos b cos c + sin b sin c cos A be differentiated, all the parts being variable; we find sin a da = (sin b cos ccos b sin c cos A) db + sin b sin c sin Ad A Dividing by sin a, this becomes, by (7) and (3), da = cos Cdb+ cos Bdc+ sin b sin C dA (285) and in the same manner from the 2d and 3d equations of (4) we find db = cos A dc + cos Cda + sin c sin A d B (286) dc cos B da + cos Adb + sin a sin Bd C (287) From these three equations, any three of the six differentials da, db, dc, d A, d B, d C, being given, the other three may be determined by the usual processes of elimination. If any one of the parts be supposed constant, its differential will become zero, and these equations will assume simpler forms. If two of the parts be supposed constant, we can easily deduce all the equations of Arts. 145, 146, 147 and 148. CHAPTER VII. APPROXIMATE SOLUTION OF SPHERICAL TRIANGLES IN CERTAIN CASES. 154. WHEN Some of the parts of the triangle are small, or nearly 90°, or nearly 180°, approximate solutions may be employed with advantage. These are generally found by means of series. 155. In a spherical right triangle (the right angle being C), given A and c, to find b. We have tan bcos A tan c (288) which is of the form in Pl. Trig. (493), and may therefore be developed by (495) and (496) by putting x = b, y = c, p = cos A, whence If A is small, cos A is nearly equal to unity, and b exceeds c by a small quantity which is approximately found by one or more terms of the series (289). If A is nearly 180°, or cos A nearly = - 1, b exceeds - c by a small quantity, which is found by (290). For examples of the mode of computation, see Pl. Trig. Art. 255. 156. Although these solutions are termed approximate, it must not be inferred that they are less accurate in practice than the direct solution of (288) by the tables; for the logarithmic tables are themselves only approximate, and the neglect of the higher powers in such series as (289) and (290) may involve a less theoretical error than the similar neglect of the higher powers in the series by which the tables are computed. In the examples of Pl. Trig. Art. 255, the thousandths of a second were found with accuracy, which could not have been effected by a direct solution with less than eight decimal places in the logarithms. These considerations lead to the frequent employment of approximate solutions in astronomy. 157. If A and b are given, to find c, we have tan csec A tan b which is reduced to Pl. Trig. (493), by putting x = c, y = b, p = sec A, 158. Similar solutions apply to the equations of right triangles, tan a sin b tan A cot B cos c tan A the last being solved under the form tan (90°-B) = cos c tan A We may also compute, in the same manner, the auxiliaries and 9 in (122) an (134), so frequently employed in the solutions of oblique triangles. 159. In a right spherical triangle, given c and A, to find a, when A is nearly 90°. We have from which we deduce sin asin A sin c A) tan (c+ a) (293) (294) tan ≥ (c — a) = tan2 (45° . From this we may find c a, which is supposed very small, by successive approximations. For a first approximation, let a = c in the second member, and find thence the value of c -a and of a; for a second approximation substitute in the second member the value of a just found; and so on until two successive values agree as nearly as may be desired. EXAMPLE. Given A = 89°, c = 87°; find a. Here 45° — A = 0° 30′, and for the first approximation (c + a) = 87°. The direct solution of (293) gives a = 86° 50′ 16′′, but cannot give the fractions of a second without tables of more than seven figure logs. We have given this problem, however, not so much on account of its particular utility, as for the purpose of introducing the method of approximation to which it leads, and which is often employed. The process here explained may obviously be applied to any equation of the form sin x m sin y when m is nearly equal to unity. 160. In a spherical oblique triangle, given two sides and the included angle, to find the other angles and side by series. If a, b and C are the data, to find c, we have tana which is of the form Pl. Trig. (507), and may be developed by (508) by substituting sin ≥ c for c, sina cos b for a, and cos 1⁄2 a sin 1⁄2 b for b; so that (508) becomes log sin c = log cos } a sin § 6 - Mtan Comparing these equations with Pl. Trig. (493), and developing by (495), we find from which a selection will be made in any particular case, according to the convergency of the series. The terms of the series are in arc, and must be reduced to seconds, by dividing by sin 1". This solution may be applied to the case where two angles and the included side are the data, by means of the polar triangle. 161. To express the area of a spherical triangle in series. Comparing (229) with Pl. Trig. (500), and developing by (502), we find K = tana tan 6 sin C — tan2 a tan2 6 sin 2 C + &c. (300) |
