BOOK V. SPHERICAL TRIGONOMETRY. DEFINITIONS. 145. SPHERICAL TRIGONOMETRY treats of methods of computing spherical triangles. 146. A SPHERICAL TRIANGLE is a portion of the surface of a sphere bounded by three arcs of a great circle, each of which is less than a semi-circumference. The three planes in which the arcs lie form a polyedral angle at the centre of the sphere. The ANGLES of a spherical triangle are the diedral angles made by the plane faces which form the polyedral angle. 147. The sides and angles of spherical triangles are usually both expressed in degrees, minutes, &c. The circumference, however, is sometimes supposed to be divided into 24 equal parts, called hours; each hour into 60 equal parts, called minutes of time; each minute into 60 equal parts, called seconds of time. Then a side is expressed by the number of hours, minutes, seconds, and decimal parts of a second, which it contains. Hours, minutes, and seconds are denoted by h., m., and s. Thus, 3h. 35m. 5.8s. RELATIONS BETWEEN THE SIDES AND ANGLES OF SPHERICAL TRIANGLES. 148. In any spherical triangle, the sines of the sides are proportional to the sines of the opposite angles. DA', DC', perpendicular to OA, O C, respectively; join B'A', B' C'. B'C' O is a right angle (Geom., Prop. VI. Bk. VII.); therefore, B COB' sin B' O CO B' sin a, and B D = B'C' sin B'C' D = B'C' sin C In like manner, BD= O B' sin c sin A; and, by the two preceding equations, O B' sin a sin C = 0 B' sin c sin A, The figure supposes a, c, B, C, &c. each less than 90°, but the relation stated may be shown to hold when the figure is modified to meet any case whatever. For instance, if C alone is greater than 90°, the point D will fall beyond O C, instead of between OC and OA; then, BCD will be the supplement of C, and thus, since the sine of an angle and its supplement are the same, the sine of BCD is still equal to the sine of C. 149. In any spherical triangle, the cosine of any side is equal to the product of the cosines of the other two sides, plus the product of the sines of those two sides into the cosine of their included angle. Let ABC be any spherical triangle, O the centre of the sphere. B Draw the plane B'A' C' perpendicular to O A. B a C' Then the an gle B'A'C' is equal to the angle A, the angle B'OC' measures the side a, and in the triangles A'B'C', O B'C' we have, by Art. 113, 2 2 2 Α' A BC= AB+ A C-2 A' B'X A' C cos A, 2 2 B' C = 0 B' + O C 2 2 0 B' XO C' cos a. Subtracting the first equation from the second, observing that C O B-A' B' and 0 C-AC are each equal to O A', since the triangles A' B', O A' C' are right-angled at A', we have 2 0=20A + 2 A' B'× A' C' cos A 2 0 BX OC" cos a; therefore, COS a + O A' X O A' A' B'X A' C' cos A. Substituting the functions derived from the triangles O A' B', cos c = cos a cos b+sin a sin b cos C. The preceding construction supposes the sides b and c, which contain the angle A, to be both less than 90°, but the formulæ obtained may be shown to be applicable in all cases. 150. In any spherical triangle, the cosine of any angle is equal to the product of the sines of the other two angles into the cos a' = cos b' cos c' + sin b' sin c' cos A'; or, by (46), COS A= whence, cos Asin B sin C cos a-cos B cos C. (153). 151. In any spherical triangle, the cotangent of one side into the sine of another side is equal to the cotangent of the angle opposite the first side into the sine of the included angle, plus the cosine of the second side into the cosine of the included angle. By (150) and (152) we have cos acos b cos c + sin b sin c cos A, cos c = cos a cos b + sin a sin b cos C'; Substituting these values of cos c and sine in the first equation, we obtain cos a= (cosa cos b+sina sin b cos C') cos b+ or, sin a sin b cos A sin C sin 4 cos a = cos a cos2b+ sin a sin b cos b cos C+ sin a sinb cot A sin C. Therefore, transposing cos a cos2b, and observing that, by (11), cos acos a cos2 b = cos a sin2 b, we have cos a sin2 b = sin a sin b cot A sin C+ sin a sin b cos b cos C, and dividing the whole by sin a sin b, we obtain cot a sin b = cot A sin C+ cos b cos C. (156) 152. By interchanging the letters in (156), we obtain (157) (158) (159) (160) (161) 153. The formule developed in the preceding articles are general, and apply to every case of spherical triangles, but require some transformations to render them more convenient for logarithmic computations. The formula (150), (151), and (152) of Art. 149 are considered the fundamental formulæ of spherical trigonometry, since from them all its other formulæ may be deduced. 154. To express the sine, cosine, and tangent of half an angle of a triangle as functions of the sides. By means of (150) we have cos a cos b cos c cos A = sin b sin c but this formula is not suited to logarithmic computation. (162) We then subtract each member of the equation from 1, and obtain (Art. 63), |