TABLE for determining the affections of the sides and angles found by the preceding analogies. I. AC and B are of the same affection (35). If BC < 90°, AB and B are of the same affection; otherwise different (36). 1 2 If BC 90°, C and B are of the same affection; otherwise different (36). 3 4 M. AB and C are of the same affection (35). If AC and C are of the same affection, BC < 90°; otherwise, BC > 90° (36). B and AC are of the same affection (35). III. Ambiguous (117). Ambiguous. Ambiguous. IV. When BC < 90°, AB and AC are of the same affection; otherwise of different affections (36). AC and B are of the same affection. When BC 90°, AC and C are of the same affection; otherwise of different affections (36). V. BC < 90°, when AB and AC are of the same affection (36). B and AC are of the same affection. C and AB are of the same affection. VI. AB and C are of the same affection. AC and B are of the same affection. When B. and C are of the same affection, BC < 90°; otherwise, BC > 90° (36). 121. Remark. In plane trigonometry, if the three angles of a triangle be given, the ratio of the sides may be found, but not their absolute values. In spherical trigonometry, if the three angles of a triangle be given, the sides can be determined from them. Another remarkable difference subsists between plane and spherical triangles. In the former, when two angles are known, the third angle also is given; but in the latter, all the angles are independent of one another, and therefore must be found separately. Solution of the Cases of Oblique-angled Spherical Triangles. 122. In any oblique-angled spherical triangle, of the three sides and the three angles any three being given, the other three may be found. The different cases or varieties which may happen in the solution of oblique-angled spherical triangles are twelve. But these varieties may be reduced to six, if they be restricted to such as depend upon the same principles for their solution. Given, to find the other parts, 1. Two sides and an angle opposite to one of them, 2. Two angles and a side opposite to one of them, 3. Two sides and the included angle, 4. Two angles and the side between them, 5. The three sides, 6. The three angles. 123. If a perpendicular arc be drawn from one of the angles upon the opposite side, all the cases of oblique-angled triangles, except the two where the three sides or the three angles are given, may be resolved by means of the preceding analogies for the resolution of right-angled triangles. But all the cases of oblique-angled spherical triangles may be resolved without a perpendicular, by means of the following theorems, which are better adapted to practice, and less difficult to remember, than the various particulars which must be attended to in the first method, with respect to the falling of the perpendicular, and the species of the different parts of the triangle. 1. The sine of any side : sine of its opposite angle :: sine of any other side : sine of its opposite angle. 2. The sine sum of any two sides sine their difference :: cot.included angle tan. difference of the other two angles. 3. The cos.sum of any two sides cos. their difference :: cot. : tan. 4. The sine sine : tan. included angle sum of the other two angles. sum of any two angles their difference :: tan.included side difference of the other two sides. 5. The cos.sum of any two angles cos.their difference :: tan.included side tan.sum of the other two sides. 6. The rectangle under the sines of any two sides : R2 :: sine sum of the three sides x sine dif. between the half sum and the third side : cos.2 included angle. 7. The rectangle under the sines of any two angles : R2 :: cos. sum of the three angles x cos. dif. between the half sum and the third angle All these theorems are adapted to logarithmic computation, and are applicable to any spherical triangle. Theorems 2, 3, 4, 5 may be varied by inversion, or alternation. Thus, theorem 2 becomes, by inversion, sine difference of two sides : sine their sum :: tan. difference of two angles cot.angle included between the two sides. P This analogy finds the third angle, when the two sides including it, and the other two angles are known. 124. In the resolution of spherical triangles there is a source of error which, in some instances, ought not to be neglected. The error arises from the magnitude of an arc, and the trigonometrical term by which its value is expressed. Thus, if a great arc be expressed by a sine, or a small arc by a cosine, there may be an error in its value found by the solution. An arc between 45° and 135° is commonly called a great arc; and an arc under 45°, or above 135°, is usually called a small arc. The former is sometimes called a mean arc, and the latter an extreme arc. But the error in a solution is of no consequence, unless an arc be very near 90°, and its value be expressed by a sine; or very near 0°, and its value be expressed by a cosine. 125. These vague or inaccurate solutions arise from the unequal manner in which the sine of an arc increases from 0° to 90°, or the cosine decreases; for the sine of a small arc increases by great increments, and the sine of a great arc by small increments; but the cosine of a small arc decreases by small increments, and the cosine of a great arc by great increments. A very small arc and its sine are very nearly equal; therefore the sine will increase nearly as fast as the arc. The sine of a great arc is nearly perpendicular to the arc; therefore it increases very little by a small change of its situation. Hence the sines near 90° increase by small increments, while those near 0° increase by great increments. The contrary is the case with respect to the cosines. The unequal variation of the sines and cosines of arcs is observable in the log. tables. The log. difference between the sines of the seconds is great near 0°, but very small near 90°, where there is no difference of the sines for nearly 100 seconds. 126. Hence it is manifest that, if a great arc be expressed by a sine, or a small arc by a cosine, its value cannot be determined with certainty. To avoid such a doubtful solution the following rule must be observed. If an arc be great its value must not be expressed by a sine; and if an arc be very small, its value must not be expressed by a cosine, The log. differences of the tangents and cotangents are considerable in all parts of an arc; therefore if the value of any arc be expressed by a tangent or a cotangent, no error will arise in the solution of a problem. Hence, if any vague solution be likely to occur from a sine or a cosine, the denomination must be changed to a tangent or a cotangent; that is, the value of a sine or a cosine must be expressed by a tangent or a cotangent, and be substituted for the former in the result. 127. Numerical Solution of the Cases of Right-angled Case 1. Given the hypothenuse BC 64° 40′, and the angle C 48°, to find the rest. AB is less than 90°, because its op. angle C is acute (35). AC is less than 90°, because BC and C are like (37). B is acute, because BC and C are like (38). Case 2. Given the side AC 54° 43', and its adjacent angle C 48°, to find the rest. AB is less than 90°, because its op. angle C is acute (35). |