[451] Similarly [452] [45] cos a = cos b cos c + sin á sin c cos a. cos b = cos a cos c + sin a sin c cos ẞ. In Fig. 124, both b and c are less than 90°, while no restriction is placed upon a or a. The resulting formulas are true, however, in general as may easily be shown. In Fig. 125, let ABC be a spherical triangle with c > 90° and b < 90°. Complete the great circle arcs to form the triangle DCA, in which AD (180° - c) < 90°. 180°c, 180° — a, 180° — a, B, and b. = - The parts of DCA are Then by [451], cos (180°-a) = cos bcos (180°-c) + sinb sin (180°-c) cos (180°-a). .. cos a = cos b cos c + sin b sin c cos a. Exercise. Draw a spherical triangle in which the two sides b and c are each greater than 90°, and verify formula [451]. 123. Theorem. The cosine of any angle of a spherical triangle is equal to the product of the sines of the other two angles multiplied by the cosine of their included side, diminished by the product of the cosines of the other two angles. Let ABC be the spherical triangle of which A'B'C' is the polar triangle. Then a 180° a', b = 180° - B', c = 180° — y', and = 180° - a'. απ Substituting these values in [451] and simplifying, This formula expresses a relation between the parts of a polar triangle. But the relation is true for any triangle since for every spherical triangle there is a polar triangle and conversely. (a) Subtracting each member of (1) from unity, rc COS α = — 2 sin (b-c+ a) sin 1⁄2 (b − c − a), - sin (ab+c). 1 sin (a+bc) sin (a - b+c) απ 1 sin b sin c a+b+c=2 s. a + b C = 2 (sc), a α = sina b+c=2 (sb). sin (8 (b) By adding each member of (1) to unity, and carrying out the work in a similar manner to that in (a), there are obtained the following: 125. Given the three angles to find the sides. If in the formulas [47] and [48] the parts of the spherical triangle be replaced by their values in terms of the parts of the polar triangle, the following formulas are obtained, where S = (a + B + y): / Y) a) cos (S -- Y). sin a sin y a) cos (S sin a sin B cos S cos (S -- sin ẞ sin y sin a sin y cos S cos (S sin a sin ẞ Y) B) Note. Since 90° <S < 270°, cos S is negative. Also, since, in the polar triangle, any side is less than the sum of the other two, Similarly it can be shown that cos (S-8) and cos (Sy) are each positive. This makes the radical expressions of this article real. Further, the positive sign must be given to the radicals in each case, for a, b, and c are each less than 90°. 、 Replacing a, b, c, a, and ẞ by their values in terms of the parts of the polar triangle, [53] becomes Replacing a, b, c, a, and 8 by their values in terms of the parts of the polar triangle, [55] becomes Equations [53], [54], [55], and [56] are known as Napier's analogies. By making the proper changes in a, b, c, a, B, and y, the corresponding formulas may be written for the other parts of the triangle. |