SPHERICAL TRIGONOMETRY CHAPTER VIII FUNDAMENTAL RELATIONS 69. Spherical trigonometry deals with the relations among the sides and angles of a spherical triangle; that is, of a portion of the surface of a sphere bounded by the intersecting arcs of three great circles. It deals also with the computation of unknown parts of such a triangle from parts which are known, the process being called, as in plane trigonometry, the solution of the triangle. The sides of a spherical triangle, being arcs of circles, are expressed in degrees, minutes, and seconds, and, as is customary, we shall consider only those triangles in which each part (angle or side) is less than one hundred and eighty degrees. 70. Law of Cosines. There is one theorem, the law of cosines, which may be called the fundamental theorem of spherical trigonometry, because by means of the theorem any spherical triangle may be solved when three of its parts are known. We shall proceed to prove the Law of Cosines. Let ABC, Fig. 32, be a spherical triangle on a sphere whose center is O, and let the sides b and c be less than 90°. 94 Through any point, A', on OA pass a plane perpendicular to OA cutting the planes OAC, OAB, and OBC, in A'C', A'B', and B'C', respectively. Then the angle B'A'C' is the measure of the diedral angle B-OA-C and, therefore, of the spherical angle A. Also, by the construction, the angles OA'B' and OA'C' are right angles. In the triangle A'B'C' B'C12 = B'A'2 + Ỡ'A'2 — 2 B'A' · C'A' cos A, — B'02 — B'A'2 + CO2 — C'A2+2 B'A'. C'A' cos A. But B'OA' and C'OA' are right triangles, and therefore, B'02 — B'A'2 = 0A"; CO2 — C'A2 – OA"2. We then have or 12 = B'O • C'O cos a = OA12 + B'A' · C'A' cos A, OA' OA' B'A' C'A' cos α = + cos A. In the above demonstration the sides b and c were taken less than 90° in order that the construction of the right triangles B'OA' and C'OA' might be possible. The resulting theorem, however, is true in all cases. Let us assume 90°<b<180° and 90° <c< 180°. Then, Fig. 33, produce the arcs AB and AC to meet in A', thus forming a lune. In the triangle A'BC, b' and c' are less than 90°. The law of cosines is, therefore, true for the triangle A'BC, so that, since A'= A, Produce the arcs forming a lune. Again, let b < 90° and 90° <c< 180°. BA and BC, Fig. 34, to meet in B', thus Then, in the triangle AB'C, b < 90° and c'< 90°, and, therefore, But cos a' = cos b cos c' + sin b sin c' cos CAB'. a' = 180° — a, c'= 180° - c, and CAB' = 180° — A. Hence cos (180° — a) |