GENERAL OBSERVATIONS ON THE SPECIES AND AMBIGUITY OF THE CASES. (A) The species of the sides and angles may be determined from the equations produced by Baron Napier's Rules, or from the preceding formulae, by attending to the signs of the quantities which compose the equations or formulae. The sides which contain the right angle are each of the same species as their opposite angles, viz. a is of the same species with A, and b is of the same species with B. (R. 145.) It may be proper to observe that where a quantity is to be determined by the sines only, and a side or angle opposite to the quantity sought does not enter into the equation, the case will be ambiguous, thus in the XIIth case, where sine b = sine a . r - - sine A sine A. sine b • Again, in the IId case, where sine a of a is evidently determinate, because it is of the same species with A which is a given quantity. (B) When an unknown quantity is to be determined by its cosine, tangent, or cotangent, the sign of this value will always determine its species; for, if its proper sign be-H, the arc will be less than 90°; if the proper sign be-, the arc will be greater than 90°. (K. 100.) (C) Again, in Case with, where radix cos b-cos c x cos a, it is obvious that the three sides are each less than 90°, or that two of them are greater than 90°, and the third less; as no other combination can render the sign of cos c x cos a like that of cos b as the equation requires.” QUADRANTAL TRIANGLES. (D) Any spherical triangle of which A, B, C, are the angles, and a, b, c, the opposite sides, may be changed into a spherical triangle of which the angles are supplements of the sides a, b, c, * Legendre's Geometry, 6th Edition, page 381. and the sides supplements of the angles A, B, C, (U 137.) viz. SOLUTIONs of THE DIFFERENT CASEs of OBLIQUE-ANGLED CASE I. Given two sides of an oblique-angled spherical triangle, and an angle opposite to one of them, to find the angle opposite to the other. CASE II. Given two sides of an oblique-angled spherical triangle, and an angle opposite to one of them, to find the angle contained between these sides. . SoLUTION. Find the angle opposite to the other given side A. Case III. Given two sides of an oblique-angled spherical triangle, and an angle opposite to one of them, to find the other side. . Solution. Find the angle opposite to the other given side Also, I. Tang p = cos C. tanga II. Tang p tang CASE IV. Given two angles of an oblique-angled spherical triangle, and a side opposite to one of them, to find the other opposite side. SoLUTION. Sinea- - Sine b sine A. sine b sine A. sine c sine C sine B. sine a sine B. sine c sine A Sine c sine C CASE W. Given two angles of an oblique-angled spherical triangle, and a side opposite to one of them, to find the side adjacent to these angles. CASE VI. Given two angles of an oblique-angled spherical triangle, and a side opposite to one of them, to find the third angle. SoLUTION. Find the side opposite to the other given angle by Case IV. cos B. sine Case VII. Given two sides of an oblique-angled spherical triangle, and the angle contained between them, to find the other angles. Here b, c, and the included ZA are given, and formulae will be obtained by a mere change of letters, if a, b, and the Z. C.; or if a, c, and the Z B are given. CASE VIII. Given two sides of an oblique-angled spherical triangle, and the angle contained between them, to find the third side. two angles by Case VII, and then find the third side by Case V. Or, the sides a, b, or crespectively, may be found by the fourth set of equations, page 184. |